Chapter 11 Conic Sections (Concepts)

Sections of a Cone

The study of conic sections begins with the geometric construction of a three-dimensional surface known as a double-napped right circular cone. This surface serves as the parent figure from which various two-dimensional curves like circles, ellipses, parabolas, and hyperbolas are derived through planar intersections.

In analytical geometry, a cone is not merely a solid object but a surface generated by the rotation of a line. Understanding the components that form this cone is essential for visualizing how different angles of a slicing plane create distinct conic shapes.

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Formation of a Double-Napped Right Circular Cone

A double-napped right circular cone is generated by the rotation of a straight line about a fixed axis while maintaining a constant angle. This process creates a hollow, symmetrical surface that extends infinitely in both directions from a central point.

The Geometric Construction Process

To visualize the formation, consider a fixed vertical line $l$ (the axis) and another line $m$ (the generator) that intersects line $l$ at a fixed point $V$ (the vertex). Let the angle between these two lines be $\alpha$.

When the line $m$ is rotated around the line $l$ such that the angle $\alpha$ remains constant, the resulting path traced by the line $m$ in three-dimensional space forms the surface of the cone. Because the line $m$ extends infinitely on both sides of the vertex $V$, the rotation produces two identical conical shapes joined at their tips.

Key Components and Terminology

The following table summarizes the fundamental elements involved in the construction of a right circular cone:

Component	Symbol	Definition and Role
Vertex	$V$	The fixed point of intersection where the axis and generator meet. It is the "pivot" of the cone.
Axis	$l$	The fixed straight line in space about which the generator rotates. It represents the line of symmetry.
Generator	$m$	The moving line that rotates around the axis to create the surface of the cone.
Semi-vertical Angle	$\alpha$	The fixed acute angle ($0 < \alpha < 90^\circ$) between the axis and the generator.

The Concept of Nappes

The vertex $V$ divides the entire surface of the cone into two distinct, infinitely extending parts. These two parts are referred to as nappes:

1. Upper Nappe: The portion of the cone extending above the vertex $V$.

2. Lower Nappe: The portion of the cone extending below the vertex $V$.

In standard conic sections, a plane might intersect only one nappe (as in a circle, ellipse, or parabola) or both nappes simultaneously (as in a hyperbola).

Mathematical Properties of the Surface

The term "Right Circular" is used because the cross-section of the cone perpendicular to the axis $l$ is always a circle. The term "Double-Napped" highlights that the surface exists on both sides of the vertex.

If we consider the axis $l$ to be the z-axis and the vertex $V$ at the origin $(0, 0, 0)$, the relationship between any point $(x, y, z)$ on the surface of the cone and the semi-vertical angle $\alpha$ can be expressed by the following equation:

This equation proves that for a fixed height $z$, the equation $x^2 + y^2 = r^2$ describes a circle with radius $r = |z| \tan \alpha$, which increases linearly as we move further away from the vertex $V$.

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Intersection of a Plane and a Cone

The study of conic sections is fundamentally the study of how a plane interacts with a double-napped right circular cone. The resulting intersection curve depends entirely on the orientation of the slicing plane. To classify these curves, we compare the semi-vertical angle ($\alpha$) of the cone with the inclination angle ($\beta$) of the plane relative to the axis of the cone.

Degenerate Conic Sections

In the study of analytic geometry, Degenerate Conic Sections refer to the singular cases where the intersecting plane passes directly through the vertex ($V$) of the double-napped cone. While standard conics result in smooth curves (circles, ellipses, parabolas, and hyperbolas), degenerate conics "collapse" into simpler geometric objects like points or lines. These are considered the limiting states of the standard conic curves.

The type of degenerate conic formed depends strictly on the comparison between the cone's semi-vertical angle ($\alpha$) and the plane's inclination angle ($\beta$) relative to the axis of the cone.

Case I: A Single Point (Degenerate Circle or Ellipse)

This occurs when the intersecting plane passes through the vertex $V$ and its inclination angle $\beta$ satisfies the condition $\alpha < \beta \le 90^\circ$.

In this scenario, the plane is relatively "flat" compared to the steepness of the cone's generator. Because it passes through the vertex but is not steep enough to cut into the nappes, the plane only makes contact with the cone at one singular point—the Vertex. This is the limiting case of an ellipse as its major and minor axes shrink to zero length.

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Case II: A Single Straight Line (Degenerate Parabola)

This occurs when the plane passes through the vertex $V$ and the inclination angle $\beta$ is exactly equal to the semi-vertical angle $\alpha$ ($\beta = \alpha$).

Under this specific condition, the plane is tangent to the surface of the cone. It lies perfectly along one of the generator lines of the cone. Since the plane passes through the vertex and remains in contact with the surface along its entire length, the resulting intersection is a single straight line. This represents the boundary state of a parabola as it narrows infinitely.

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Case III: A Pair of Intersecting Lines (Degenerate Hyperbola)

This occurs when the plane passes through the vertex $V$ and is steeper than the generator line, satisfying $0 \le \beta < \alpha$.

In this orientation, the plane is steep enough to slice through both the upper and lower nappes of the cone. Because the plane passes through the vertex, the intersection consists of two generator lines that cross each other at the vertex $V$, forming an "X" shape. This is a degenerate hyperbola, representing the state where the two branches of a hyperbola finally meet at their center point.

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Summary Table of Degenerate Conics

The following table provides a quick reference for identifying the geometric result when a plane passes through the vertex of a cone:

Relationship of Angles	Geometric Result	Standard Conic Equivalent
$\alpha < \beta \le 90^\circ$	A Point	Ellipse / Circle
$\beta = \alpha$	A Single Straight Line	Parabola
$0 \le \beta < \alpha$	Two Intersecting Lines	Hyperbola

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Standard (Non-Degenerate) Conic Sections

Standard or Non-Degenerate Conic Sections are the smooth curves formed when an intersecting plane slices through a double-napped right circular cone without passing through its vertex. The shape of the resulting curve is entirely dependent on the relationship between the cone's semi-vertical angle ($\alpha$) and the angle ($\beta$) that the plane makes with the axis of the cone.

1. The Circle ($\beta = 90^\circ$)

A circle is generated when the intersecting plane is perfectly perpendicular to the axis of the cone. In this specific case, the angle $\beta$ is exactly $90^\circ$.

Because the plane cuts horizontally across a single nappe, every point on the resulting curve is equidistant from the axis. Geometrically, a circle is a special version of an ellipse where the slicing is not slanted. It is the most symmetrical of all conic sections.

2. The Ellipse ($\alpha < \beta < 90^\circ$)

An ellipse is formed when the plane is tilted at an angle $\beta$ that is greater than the semi-vertical angle $\alpha$ but less than $90^\circ$.

In this range of angles, the plane is slanted but not steep enough to become parallel to the generator or to hit the second nappe. It cuts completely through one nappe, resulting in a closed, oval-shaped curve. The smaller the angle $\beta$ (as it approaches $\alpha$), the more elongated or "stretched" the ellipse becomes.

3. The Parabola ($\beta = \alpha$)

A parabola occurs at the precise moment when the inclination of the plane matches the inclination of the cone's generator line, meaning $\beta = \alpha$.

In this orientation, the plane is parallel to the generator. Because it is parallel, the plane can never "exit" the other side of the nappe to close the curve. The result is an open, U-shaped curve that extends infinitely in one direction. It serves as the critical transition point between the closed ellipse and the two-branched hyperbola.

4. The Hyperbola ($0 \le \beta < \alpha$)

A hyperbola is formed when the plane is steeper than the generator line, such that the angle $\beta$ is less than the semi-vertical angle $\alpha$.

Because the plane is so steep, it is guaranteed to slice through both the upper and lower nappes of the cone. This creates two distinct, unbounded curves that open in opposite directions, known as the "branches" of the hyperbola. If the plane is perfectly parallel to the axis ($\beta = 0^\circ$), the hyperbola is symmetric across the vertex.

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Summary of Formation Conditions

The following table provides a concise comparison of the conditions required for each standard conic section. Understanding these inequalities is crucial for conceptual clarity in coordinate geometry.

Conic Section	Plane Angle ($\beta$)	Nature of Curve	Nappes Intersected
Circle	$\beta = 90^\circ$	Closed & Symmetrical	One
Ellipse	$\alpha < \beta < 90^\circ$	Closed Oval	One
Parabola	$\beta = \alpha$	Open (Infinite)	One
Hyperbola	$0 \le \beta < \alpha$	Two Open Branches	Two

Circle and its Various Forms

In the branch of coordinate geometry, a circle is defined as the locus of a point that moves in a 2D plane such that its distance from a fixed point remains constant. This geometric figure is one of the most fundamental conic sections, formed when a plane cuts a right circular cone perpendicular to its axis.

The fixed point in the plane is known as the centre of the circle, and the constant distance from this centre to any point on the boundary is called the radius. It is a fundamental property that the radius ($r$) must always be a positive real number ($r > 0$).

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1. Standard (or Simplest) Form

The Standard Form (also frequently referred to as the Simplest Form) of the equation of a circle occurs when the centre of the circle coincides with the origin of the Cartesian coordinate system. This specific orientation simplifies the algebraic expression significantly as the coordinates of the centre become zero.

Derivation of the Equation

Let the origin $O(0, 0)$ be the fixed centre of the circle and let $r$ be the fixed radius. We consider an arbitrary point $P(x, y)$ that lies on the circumference of the circle.

According to the geometric definition of a circle, the distance between the centre $O$ and the point $P$ must be equal to the radius $r$:

Using the Distance Formula between two points $(x_1, y_1)$ and $(x_2, y_2)$, given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, we can write the distance $OP$ as:

Simplifying the expression inside the square root:

To obtain a polynomial equation, we square both sides of the equation:

This yields the final Standard Equation of a Circle:

This equation states that for any point $(x, y)$ on the circle, the sum of the squares of its coordinates is always equal to the square of the radius. This is also a direct application of the Pythagoras Theorem in a right-angled triangle where $x$ and $y$ are the legs and $r$ is the hypotenuse.

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2. Central Form

The Central Form of the equation of a circle, also widely known as the Center-Radius Form, is the most generalized way to represent a circle in a 2D plane. While the standard form assumes the center is at the origin, the central form allows the center to be located at any arbitrary coordinate $(h, k)$ in the Cartesian plane.

This form is highly useful in analytical geometry because it explicitly reveals the two most important geometric properties of a circle: its position (coordinates of the center) and its size (the radius).

Derivation of the Central Form

To derive this equation, we use the fundamental definition of a circle: the locus of a point moving such that its distance from a fixed point is constant.

Let $C(h, k)$ be the fixed point, which we call the center of the circle. Let $r$ be the constant distance, which is the radius ($r > 0$). Now, consider any arbitrary point $P(x, y)$ that lies on the boundary of the circle.

By the definition of a circle, the length of the line segment joining the center $C$ to the point $P$ must be equal to the radius $r$:

Using the Distance Formula to calculate the length of $CP$ between points $(h, k)$ and $(x, y)$, we have:

To simplify the equation and remove the radical (square root), we square both sides of equation (i):

This results in the Central Form of the equation of a circle:

In this equation, $(h, k)$ are the coordinates of the center and $r$ is the radius. If we know these two values, we can immediately write down the equation of the circle.

Special Observations

The Central Form is a versatile equation. Based on the values of $h$, $k$, and $r$, we can observe the following cases:

Condition	Equation	Geometric Result
$h = 0, k = 0$	$x^2 + y^2 = r^2$	Circle centered at the Origin.
$r = h$	$(x - h)^2 + (y - k)^2 = h^2$	Circle touches the y-axis.
$r = k$	$(x - h)^2 + (y - k)^2 = k^2$	Circle touches the x-axis.
$r = h = k$	$(x - r)^2 + (y - r)^2 = r^2$	Circle touches both axes.

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3. General Form of the Equation of a Circle

The General Form of the equation of a circle is a specialized case of the general second-degree equation in two variables. It is the form most commonly encountered in analytical geometry problems, especially when the center and radius are not immediately apparent. By expanding the central form, we can represent any circle in a standardized algebraic structure.

Every circle can be represented by a second-degree equation in $x$ and $y$. However, for an equation to specifically represent a circle, it must satisfy certain conditions regarding its coefficients. The general form is obtained by expanding the Central Form and regrouping the constants.

Mathematical Derivation

We begin with the central form of the equation of a circle with center $(h, k)$ and radius $r$:

Expanding the binomial terms using the identity $(a - b)^2 = a^2 - 2ab + b^2$:

$(x^2 - 2hx + h^2) + (y^2 - 2ky + k^2) = r^2$

Rearranging the terms such that the second-degree terms are together, followed by the first-degree terms and finally the constants:

$x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0$

To make this equation more standard, we introduce new constants $g$, $f$, and $c$. We define them as follows:

Substituting these into the expanded equation, we obtain the General Form:

Properties of the General Equation

The general equation of a circle is expressed as $x^2 + y^2 + 2gx + 2fy + c = 0$. This algebraic form is derived by expanding the central form $(x - h)^2 + (y - k)^2 = r^2$. To understand the geometric properties (centre and radius) hidden within this general form, we must perform a process called "completing the square".

Derivation of Centre and Radius from General Form

To find the centre and radius, we rearrange the general equation to group the $x$ and $y$ terms together:

To make the terms in the brackets perfect squares, we add $g^2$ and $f^2$ to both sides of the equation:

The left-hand side can now be written as the sum of two squares:

Comparing equation (i) with the standard central form $(x - h)^2 + (y - k)^2 = r^2$, we can extract the specific properties of the circle.

1. Finding the Centre

By comparing $(x - h)$ with $(x + g)$, we find $h = -g$. Similarly, comparing $(y - k)$ with $(y + f)$, we find $k = -f$. The centre $(h, k)$ is therefore determined by the coefficients of $x$ and $y$ as follows:

Thus, the coordinates of the Centre are always $(-g, -f)$.

2. Finding the Radius

Comparing the right-hand sides of the equations, we have $r^2 = g^2 + f^2 - c$. Since the radius must be a non-negative distance, we take the positive square root:

For the circle to exist in the real Cartesian plane, the expression under the square root must satisfy $g^2 + f^2 - c \geq 0$.

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Summary Table of Properties

The following table serves as a quick reference for identifying the properties of a circle from its general second-degree equation.

Property	Algebraic Expression	Geometric Identification
Centre $(h, k)$	$(-g, -f)$	Negative half of the coefficients of $x$ and $y$.
Radius $(r)$	$\sqrt{g^2 + f^2 - c}$	Square root of $(\text{Centre } x)^2 + (\text{Centre } y)^2 - \text{constant term}$.
Condition for Circle	$g^2 + f^2 - c > 0$	The result under the radical must be positive for a real circle.

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Nature of the Circle

The existence and type of a circle are determined by the term under the square root in the radius formula, which is $g^2 + f^2 - c$. This expression acts as a discriminant for the circle's radius. Since the radius $r = \sqrt{g^2 + f^2 - c}$ must be a real distance, we categorize the nature of the circle into three distinct cases:

Mathematical Condition	Type of Circle	Geometric Interpretation
$g^2 + f^2 - c > 0$	Real Circle	The radius is a real positive number. The circle can be drawn in the Cartesian plane with an infinite set of points.
$g^2 + f^2 - c = 0$	Point Circle	The radius is zero. The circle "collapses" into its own centre $(-g, -f)$. It is a degenerate case of a circle.
$g^2 + f^2 - c < 0$	Imaginary Circle	The radius is an imaginary number. No real points $(x, y)$ satisfy the equation; the locus is an empty set.

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Requirements for a General Equation to represent a Circle

Consider the general second-degree equation in two variables $x$ and $y$:

For the above equation to represent a circle, the following three mandatory conditions must be satisfied:

1. Equality of Quadratic Coefficients

The coefficients of $x^2$ and $y^2$ must be exactly equal, and they must not be zero ($a = b \neq 0$). If they were different, the curve would represent an ellipse or another conic section because the "stretching" would be unequal in the $x$ and $y$ directions.

2. Absence of the Product Term (xy)

There must be no $xy$ term in the equation. Mathematically, the coefficient $h$ must be equal to $0$. The presence of an $xy$ term indicates that the axes of the conic are rotated relative to the coordinate axes, which does not happen for a circle.

3. Condition for Real Radius

After dividing the entire equation by $a$ (to make the coefficients of $x^2$ and $y^2$ equal to $1$), the resulting coefficients $g, f,$ and $c$ must satisfy the inequality $g^2 + f^2 - c \geq 0$. If this is not met, the equation does not correspond to any real geometric figure.

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Shortcut Rule to find Centre and Radius

Instead of completing the square every time, you can use the following shortcut method to identify the properties of a circle from its general equation.

Step 1: Normalize the Equation

Ensure that the coefficients of $x^2$ and $y^2$ are unity (equal to $1$). If the equation is $ax^2 + ay^2 + 2gx + 2fy + c = 0$, divide every term by $a$ first:

Step 2: Apply the Half-and-Negative Rule

To find the centre, simply take the coefficients of $x$ and $y$ from the normalized equation and divide them by $-2$:

Step 3: Calculate the Radius

Once you have the centre $(h, k)$, the radius is found by squaring the coordinates of the centre, adding them, and subtracting the constant term $c$ (from the normalized equation):

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4. Diameter Form

If we are given two points $A(x_1, y_1)$ and $B(x_2, y_2)$ as the extremities (endpoints) of a diameter, we can find the equation of the circle without explicitly calculating the centre or the radius first.

Geometric Derivation

Let $A(x_1, y_1)$ and $B(x_2, y_2)$ be the endpoints of the diameter of a circle. Let $P(x, y)$ be any arbitrary point on the circumference of this circle (other than $A$ or $B$).

According to the theorem of Thales, the angle $\angle APB$ is a right angle ($90^\circ$). This implies that the line segment $AP$ is perpendicular to the line segment $BP$.

Now, let $m_1$ be the slope of the line $AP$ and $m_2$ be the slope of the line $BP$. Using the coordinates of the points:

Since the two lines are perpendicular, the product of their slopes must be $-1$:

Substituting the expressions for $m_1$ and $m_2$ into equation (i):

Multiplying the numerators and denominators:

Cross-multiplying to eliminate the fraction:

Rearranging all terms to one side, we get the Diameter Form of the equation of a circle:

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Summary Table of Circle Forms

Depending on the geometric data provided—such as the center, radius, or endpoints of a diameter—we use different forms of the circle equation. The following table provides a comprehensive summary of these forms:

Form Name	Conditions / Parameters Given	Standard Equation
Standard Form	Centre is at the origin $(0, 0)$ and radius is $r$.	$x^2 + y^2 = r^2$
Central Form	Centre is at $(h, k)$ and radius is $r$.	$(x - h)^2 + (y - k)^2 = r^2$
General Form	Coefficients $g, f$ and constant $c$ are known.	$x^2 + y^2 + 2gx + 2fy + c = 0$
Diameter Form	Endpoints of a diameter are $(x_1, y_1)$ and $(x_2, y_2)$.	$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$

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Geometrical Condition for the Intersection of a Line and a Circle

In analytical geometry, the interaction between a straight line and a circle is determined by their relative positions in the Cartesian plane. By comparing the perpendicular distance from the center of the circle to the line with the radius of the circle, we can precisely categorize the nature of their intersection.

Consider a circle $S$ with its center at $C(x_1, y_1)$ and a radius $r$ ($r > 0$). Let the equation of the line $l$ be given by $ax + by + c = 0$. The first step in any such geometric analysis is to calculate the perpendicular distance $d$ from the center $C$ to the line $l$.

The formula for the perpendicular distance $d$ is:

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The Three Geometrical Cases

Depending on the value of $d$ relative to $r$, we have the following three scenarios:

Case I: The Line Intersects the Circle in Two Distinct Points (Secant Line)

If the perpendicular distance $d$ from the center to the line is less than the radius ($d < r$), the line $l$ passes through the interior of the circle. In this case, the line is called a Secant.

The length of the segment $AB$ formed by the intersection is known as the length of intercept. The length of this chord is calculated as:

Case II: The Line Touches the Circle at a Unique Point (Tangent Line)

If the perpendicular distance $d$ from the center to the line is exactly equal to the radius ($d = r$), the line $l$ meets the circle at one and only one point. In this case, the line is a Tangent to the circle.

The unique point of contact $M$ is the foot of the perpendicular from center $C$ to the line $l$. A line touches a circle if and only if the length of the perpendicular from the center is equal to the radius.

Case III: The Line Does Not Intersect the Circle

If the perpendicular distance $d$ from the center to the line is greater than the radius ($d > r$), the line $l$ lies entirely outside the circle. There are no real points of intersection.

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Summary Table of Conditions

Distance Condition	Interaction Nature	Number of Common Points
$d < r$	Secant (Intersects)	$2$
$d = r$	Tangent (Touches)	$1$
$d > r$	Does not meet	$0$

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Alternative Algebraic Method

While the geometrical method (using $d$ and $r$) is preferred for its simplicity, one can also use an algebraic approach by substituting the line equation $y = mx + c$ into the circle equation. This results in a quadratic equation in $x$:

The nature of intersection depends on the Discriminant ($D$) of this quadratic equation:

• If $D > 0$, the line is a secant (two points).
• If $D = 0$, the line is a tangent (one point).
• If $D < 0$, the line does not intersect.

Relative Position of Two Circles

In coordinate geometry, the spatial relationship between two circles in a two-dimensional plane is determined by comparing the distance between their centers with the sum or difference of their radii. This analysis is fundamental for solving problems involving tangency, intersection, and common tangents.

Let us consider two non-concentric circles $S_1$ and $S_2$ with the following parameters:

• First Circle ($S_1$): Centre $A(x_1, y_1)$ and Radius $r_1$.
• Second Circle ($S_2$): Centre $B(x_2, y_2)$ and Radius $r_2$.
• Distance ($d$): The distance between the centres $A$ and $B$, calculated using the distance formula:

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Classification of Relative Positions

Based on the value of $d$ relative to $r_1$ and $r_2$, we define five distinct geometric cases.

Case I: One Circle Lies Completely Outside the Other

If the distance between the centres is greater than the sum of their radii, the circles do not touch or intersect. They are completely separate in the plane.

Case II: The Two Circles Touch Externally

When the distance between the centres is exactly equal to the sum of the radii, the circles touch at a single point $P$. This point $P$ lies on the line segment joining the centres $A$ and $B$, dividing it in the ratio $r_1 : r_2$ internally.

Case III: The Two Circles Intersect at Two Distinct Points

If the distance between the centres is less than the sum of the radii but greater than the absolute difference of the radii, the circles overlap. The two intersection points form a common chord.

Case IV: The Two Circles Touch Internally

If the distance between the centres is exactly equal to the absolute difference of the radii, one circle lies inside the other and they touch at exactly one point. The point of contact lies on the extension of the line joining the centres and divides it externally in the ratio $r_1 : r_2$.

Case V: One Circle Lies Completely Inside the Other

When the distance between the centres is less than the absolute difference of the radii, the smaller circle is entirely contained within the larger one without touching its boundary.

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Summary Table of Relative Positions

Condition	Geometric Relationship	Number of Common Tangents
$d > r_1 + r_2$	Separate (Outside)	4
$d = r_1 + r_2$	Touch Externally	3
$|r_1 - r_2| < d < r_1 + r_2$	Intersect at 2 Points	2
$d = |r_1 - r_2|$	Touch Internally	1
$d < |r_1 - r_2|$	One inside another	0

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Symmetry

Symmetry is a fundamental geometric concept that describes balance and self-similarity within a figure. In coordinate geometry, a curve is said to be symmetric if it remains unchanged after a specific transformation, such as reflection across a line or through a point. Recognizing symmetry allows us to analyze and graph complex equations by studying only a portion of the curve.

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1. Reflection of a Point

Before analyzing the symmetry of complex curves like parabolas or ellipses, we must define how an individual point $P$ behaves when reflected across a line of reflection or through a point of reflection.

Reflection in a Line (Mirror Reflection)

Consider a line $l$ acting as a plane mirror. The reflection of a point $P$ in line $l$ is a point $P'$ such that:

• If $P$ lies on the line $l$, its reflection is the point $P$ itself ($P = P'$).
• If $P$ does not lie on $l$, let $M$ be the foot of the perpendicular drawn from $P$ to the line $l$. We extend this line $PM$ to a point $P'$ on the opposite side such that the distance $PM$ is equal to $MP'$.

In this scenario, the line $l$ is the perpendicular bisector of the line segment $PP'$. This means $l \perp PP'$ and $M$ is the midpoint of $PP'$.

Reflection in a Point (Point Symmetry)

The reflection of a point $P$ in a fixed point $M$ is defined as the point $P'$ such that $M$ is the midpoint of the segment $PP'$. This is geometrically equivalent to rotating the point $P$ by $180^\circ$ around the point $M$.

• If $P$ coincides with $M$, then the reflection $P'$ is also $M$.
• If $P(x, y)$ is reflected in $M(a, b)$, the image $P'(x', y')$ is found using the midpoint formula:

Rearranging these, we get the coordinates of the reflected point $P'$:

Common Reflection Results in Coordinate Geometry

The following table summarizes the images of a point $P(\alpha, \beta)$ under standard reflections used in Indian curriculum textbooks:

Reflecting Element	Original Point	Reflected Image	Algebraic Change
x-axis ($y=0$)	$(\alpha, \beta)$	$(\alpha, -\beta)$	$y \to -y$
y-axis ($x=0$)	$(\alpha, \beta)$	$(-\alpha, \beta)$	$x \to -x$
Origin $(0, 0)$	$(\alpha, \beta)$	$(-\alpha, -\beta)$	Both signs change
Line $y = x$	$(\alpha, \beta)$	$(\beta, \alpha)$	$x, y$ are interchanged

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2. Symmetry of a Curve about a Line

In analytical geometry, determining the symmetry of a curve is a vital step in curve sketching. A curve $C$, represented by the equation $F(x, y) = 0$, is said to be symmetrical about a line $l$ if, for every point $P$ existing on the curve, its reflection image $P'$ in the line $l$ also lies on the curve. In such a case, the line $l$ is formally called the axis of symmetry of the curve.

Symmetry about the x-axis (Horizontal Symmetry)

A curve is symmetrical about the x-axis if the portion of the curve above the x-axis is an exact mirror reflection of the portion below it. This implies that for every point $(x, y)$ on the curve, the point $(x, -y)$ must also satisfy the curve's equation.

Proof and Algebraic Test

Let $P(\alpha, \beta)$ be any point on the curve $C$ defined by $F(x, y) = 0$. Since $P$ lies on the curve, its coordinates must satisfy the equation:

The reflection of the point $P(\alpha, \beta)$ in the x-axis is given by $P'(\alpha, -\beta)$. For the curve to be symmetrical about the x-axis, this reflected point $P'$ must also lie on the curve $C$. Therefore:

Comparing equations (i) and (ii), we conclude that the curve is symmetrical about the x-axis if and only if the equation remains unchanged when $y$ is replaced by $-y$.

Practical Rule: This symmetry usually exists if the variable $y$ occurs only in even powers (e.g., $y^2, y^4, \cos y$) within the equation.

Symmetry about the y-axis (Vertical Symmetry)

A curve is symmetrical about the y-axis if the left-hand side of the y-axis is a mirror image of the right-hand side. Geometrically, if a point $(x, y)$ is on the curve, then its reflection across the y-axis, $(-x, y)$, must also be on the curve.

Algebraic Test and Derivation

Following the same logic as the previous proof, the reflection of $P(\alpha, \beta)$ in the y-axis is $P'(-\alpha, \beta)$. The curve $F(x, y) = 0$ is symmetrical about the y-axis if:

This leads to the formal condition: the equation of the curve must remain unchanged when $x$ is replaced by $-x$.

Practical Rule: This symmetry usually exists if the variable $x$ occurs only in even powers (e.g., $x^2, x^4, |x|$) within the equation.

Comparison of Axial Symmetries

Feature	x-axis Symmetry	y-axis Symmetry
Point Relationship	$(x, y) \to (x, -y)$	$(x, y) \to (-x, y)$
Algebraic Requirement	Replace $y$ with $-y$	Replace $x$ with $-x$
Visual Identification	Even powers of $y$	Even powers of $x$
Standard Example	Parabola $y^2 = 4ax$	Parabola $x^2 = 4ay$

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3. Symmetry of a Curve about a Point (Origin)

A curve is said to possess point symmetry if there exists a fixed point $M$ such that for every point $P$ on the curve, its reflection through $M$ also lies on the curve. In such instances, the point $M$ is designated as the centre of symmetry or simply the centre of the curve.

The most significant case of point symmetry in coordinate geometry is symmetry with respect to the origin $(0, 0)$. Geometrically, this is equivalent to a $180^\circ$ rotation of the curve about the origin; if the rotated figure coincides perfectly with the original, the curve is symmetric about the origin.

Algebraic Test for Origin Symmetry

To determine if a curve $F(x, y) = 0$ is symmetric about the origin, we apply a simultaneous substitution. The equation must remain unchanged when $x$ is replaced by $-x$ AND $y$ is replaced by $-y$.

Formal Proof

Let $P(\alpha, \beta)$ be any arbitrary point lying on the curve $C$. Since $P$ is on the curve, it must satisfy the equation:

The reflection of the point $P(\alpha, \beta)$ in the origin $(0, 0)$ is the point $P'(-\alpha, -\beta)$. For the curve to be symmetric about the origin, $P'$ must also lie on the curve $C$, implying:

Comparing equations (i) and (ii), we find that the condition for symmetry about the origin is:

Geometric Visualization

Geometrically, if you draw a straight line from any point $P(x, y)$ on the curve through the origin, it will intersect the curve again at a point $P'(-x, -y)$ such that the origin is the midpoint of the segment $PP'$.

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Comparison of Point and Axial Symmetry

The following table distinguishes between symmetry about the axes and symmetry about the origin for a curve $F(x, y) = 0$.

Type of Symmetry	Algebraic Change	Geometric Transformation
Symmetry about x-axis	$y \to -y$	Reflection (Mirror Image)
Symmetry about y-axis	$x \to -x$	Reflection (Mirror Image)
Symmetry about Origin	$x \to -x$ AND $y \to -y$	$180^\circ$ Rotation

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Symmetry in Standard Conic Sections

Standard conic sections—the circle, parabola, ellipse, and hyperbola—possess unique symmetrical properties that are dictated by their algebraic equations. In coordinate geometry, symmetry helps in identifying the orientation of the curve and simplifies the process of finding its foci, vertices, and directrices. When a conic is in its standard form (centered at the origin or with its axis along the coordinate axes), its symmetry is most apparent.

1. The Circle

A circle represented by the equation $x^2 + y^2 = r^2$ is the most symmetrical of all conic sections. Because both $x$ and $y$ appear as even powers and the equation is symmetric with respect to both variables, it satisfies all primary symmetry tests.

Mathematical Justification

• Symmetry about x-axis: Replacing $y$ with $-y$ gives $x^2 + (-y)^2 = r^2 \implies x^2 + y^2 = r^2$. (Unchanged)
• Symmetry about y-axis: Replacing $x$ with $-x$ gives $(-x)^2 + y^2 = r^2 \implies x^2 + y^2 = r^2$. (Unchanged)
• Symmetry about Origin: Replacing both $x$ and $y$ with $-x$ and $-y$ results in the same equation.
• Symmetry about $y = x$: Interchanging $x$ and $y$ gives $y^2 + x^2 = r^2$, which is identical to the original equation.

2. The Parabola

A standard parabola, such as $y^2 = 4ax$, possesses symmetry about only one axis. In this specific form, the variable $y$ is squared, while $x$ is linear.

Mathematical Justification

If we replace $y$ with $-y$ in the equation:

However, replacing $x$ with $-x$ results in $y^2 = -4ax$, which is a different curve. Therefore, the parabola $y^2 = 4ax$ is symmetrical about the x-axis only. The x-axis is called the axis of the parabola.

3. The Ellipse

An ellipse in its standard form, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, contains even powers of both $x$ and $y$. This structure ensures multiple types of symmetry.

Mathematical Justification

Because both variables are squared, replacing $x$ with $-x$, or $y$ with $-y$, or both simultaneously, does not alter the equation. Thus, a standard ellipse is symmetric with respect to:

• The x-axis (Major or Minor axis).
• The y-axis (Minor or Major axis).
• The Origin (The center of the ellipse).

4. The Hyperbola

Similar to the ellipse, the standard hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ consists of even powers of the coordinates. Despite being an "open" curve with two branches, its algebraic symmetry is identical to that of the ellipse.

Mathematical Justification

The squared terms $\left(\frac{x}{a}\right)^2$ and $\left(\frac{y}{b}\right)^2$ ensure that the signs of $x$ and $y$ do not affect the result. Therefore, the hyperbola is symmetric about the x-axis (Transverse axis), the y-axis (Conjugate axis), and the origin.

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Summary of Symmetry in Conics

The following table summarizes the symmetry of conic sections in their standard forms:

Conic Section	Standard Equation	Axis of Symmetry	Origin Symmetry
Circle	$x^2 + y^2 = r^2$	Both Axes	Yes
Parabola	$y^2 = 4ax$	x-axis only	No
Ellipse	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	Both Axes	Yes
Hyperbola	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	Both Axes	Yes

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Parabola

In the field of analytical geometry, while a parabola is visually recognized as a specific U-shaped curve, its mathematical identity is rooted in a fundamental distance relationship. This relationship, known as the Focus-Directrix property, defines the parabola as a locus—the path traced by a point moving under specific geometric constraints.

A parabola is defined as the locus of a point $P$ in a plane such that its distance from a fixed point $S$ (the focus) is exactly equal to its perpendicular distance from a fixed straight line $l$ (the directrix).

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The Mathematical Condition: $PS = PM$

To express this property algebraically, let us consider a point $P(x, y)$ situated anywhere on the curve of the parabola. Let $S$ be the focus of the parabola and $l$ be its directrix. If we drop a perpendicular from point $P$ to the line $l$, meeting it at point $M$, then for $P$ to lie on the parabola, the following condition must be satisfied:

This equality is the "genetic code" of the parabola. If $PS < PM$, the resulting locus would be an ellipse; if $PS > PM$, it would be a hyperbola. Only the exact equality $PS = PM$ generates the parabolic curvature.

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Understanding Eccentricity ($e$)

The term Eccentricity ($e$) is a numerical measure of how much a conic section deviates from being a perfect circle. It is defined as the ratio of the distance of a point from the focus to its distance from the directrix.

Since the defining property of a parabola is $PS = PM$, substituting this into the ratio gives:

Conclusion: A parabola is the only conic section with an eccentricity equal to unity ($e = 1$).

Comparison Table of Conic Eccentricities

To place the parabola in context with other conic sections, observe the following comparison:

Conic Section	Value of Eccentricity ($e$)	Geometric Relationship
Circle	$e = 0$	Focus is at the center; directrix is at infinity.
Ellipse	$0 < e < 1$	Point is closer to the focus ($PS < PM$).
Parabola	$e = 1$	Point is equidistant from focus and directrix ($PS = PM$).
Hyperbola	$e > 1$	Point is further from the focus ($PS > PM$).

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Key Terminology of a Parabola

To perform calculations and understand the properties of a parabola, one must be well-versed with its geometric components. Each element plays a specific role in defining the orientation and width of the curve.

1. Focus (S or F)

The focus is the fixed point that, along with the directrix, defines the parabola. It does not lie on the curve itself but is located in the "interior" of the U-shape. In standard parabolas ($y^2 = 4ax$), the focus is at $(a, 0)$.

2. Directrix (l)

The directrix is the fixed straight line. The parabola opens away from this line. The distance from the vertex to the focus is always equal to the perpendicular distance from the vertex to the directrix.

3. Axis of the Parabola

The straight line passing through the focus and perpendicular to the directrix is called the axis. It serves as the line of symmetry for the curve; if the parabola is folded along its axis, the two halves will coincide perfectly.

4. Vertex (V)

The point where the parabola intersects its axis is called the vertex. It is the "turning point" of the curve. Geometrically, the vertex is the midpoint of the perpendicular segment dropped from the focus to the directrix.

5. Focal Chord and Focal Distance

Any chord passing through the focus is called a focal chord. The distance from any point $P(x, y)$ on the parabola to the focus is known as the focal distance of that point.

6. Latus Rectum (L'L)

The latus rectum is a specific focal chord that is perpendicular to the axis of the parabola. Its length is a crucial parameter as it determines how "wide" the parabola opens. For a standard parabola $y^2 = 4ax$, the length of the latus rectum is $4a$.

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Derivation of the Standard Equation ($y^2 = 4ax$)

The algebraic representation of a parabola is most efficient when the curve is placed in a "standard position." This occurs when the vertex lies at the origin $(0, 0)$ and the axis of symmetry coincides with one of the coordinate axes. The following derivation focuses on the Right-Hand Parabola, which opens along the positive x-axis.

To derive the equation, we apply the fundamental locus definition: every point $P$ on the parabola is equidistant from the focus $F$ and the directrix line $l$.

Step 1: Setting up the Coordinate System

Let $F$ be the focus and $l$ be the directrix. Draw a perpendicular from $F$ to $l$, meeting it at point $Z$. Let the distance $ZF = 2a$, where $a > 0$ is a constant. We choose the midpoint of $ZF$ as the origin $O(0, 0)$.

• Since $O$ is the midpoint and $ZF$ lies on the x-axis, the coordinates of the focus are $F(a, 0)$.
• The coordinates of $Z$ are $(-a, 0)$.
• The directrix $l$ is a vertical line passing through $Z(-a, 0)$, so its equation is $x = -a$ or $x + a = 0$.

Step 2: Applying the Locus Condition

Let $P(x, y)$ be any arbitrary point on the parabola. Let $PM$ be the perpendicular distance from $P$ to the directrix $x + a = 0$. By definition:

Using the distance formula for $PF$ and the perpendicular distance formula for $PM$:

Step 3: Algebraic Simplification

Squaring both sides of the equation to eliminate the radical:

Expanding the squared binomials:

$x^2 - 2ax + a^2 + y^2 = x^2 + 2ax + a^2$

Subtracting $x^2$ and $a^2$ from both sides:

$-2ax + y^2 = 2ax$

Rearranging the terms to isolate $y^2$:

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Detailed facts and Geometric Properties

The standard equation of a parabola, $y^2 = 4ax$ (where $a > 0$), is not just an algebraic expression but a geometric blueprint. It provides specific information about how the curve is oriented, its boundaries, and its internal dimensions.

1. Symmetry about the Axis

Symmetry is a property where one half of a figure is a mirror image of the other. For the parabola $y^2 = 4ax$, if we replace the variable $y$ with $-y$, the equation remains unchanged:

This mathematical invariance proves that for every point $(x, y)$ on the curve, there exists a corresponding point $(x, -y)$ on the curve. Geometrically, this means the x-axis (the line $y = 0$) acts as a mirror.

Conclusion: The line of symmetry for this parabola is the x-axis, which is formally designated as the Axis of the Parabola.

2. Domain, Range, and Orientation

The standard equation of a parabola, $y^2 = 4ax$ (where $a > 0$), does more than just define a shape; it establishes strict algebraic boundaries on the Cartesian plane. By analyzing the variables $x$ and $y$, we can determine exactly where the curve exists and where it is physically impossible for it to appear.

1. Algebraic Analysis of the Equation

In the real number system, the square of any value, whether positive or negative, must be non-negative. This fundamental property of squares directly limits the possible values of $x$ in the equation of the parabola.

Derivation of the Horizontal Constraint

Consider the equation $y^2 = 4ax$. Since $y^2$ represents the square of a real number:

Substituting the parabolic relation into the inequality:

Since we have defined $a$ as a positive constant ($a > 0$), we divide both sides by $4a$ without changing the inequality sign:

2. Domain and Range

Based on the algebraic derivation above, we can formally define the extent of the parabola on the coordinate axes.

The Domain

The Domain of a relation is the set of all possible input ($x$) values. As derived, $x$ cannot be negative. If we were to pick a negative value for $x$ (e.g., $x = -1$), then $y^2 = -4a$, which results in imaginary values for $y$.

• Domain: $[0, \infty)$ or $\{x \in \mathbb{R} : x \ge 0\}$.
• Geometric Meaning: No part of the parabola exists in the second or third quadrants (the region to the left of the y-axis).

The Range

The Range is the set of all resulting $y$ values. In the equation $y = \pm 2\sqrt{ax}$, as $x$ increases from zero to infinity, $y$ also increases and decreases indefinitely.

• Range: $(-\infty, \infty)$ or all real numbers $\mathbb{R}$.
• Geometric Meaning: The curve extends infinitely far upwards and infinitely far downwards.

3. Orientation: The "Right-Hand" Parabola

The combined constraints of the domain and the vertex location define the Orientation of the curve. Since the vertex is at $(0, 0)$ and the curve only exists where $x \ge 0$, the parabola must open towards the positive x-axis.

Conclusion: The parabola $y^2 = 4ax$ is oriented towards the Right Hand side. It is often referred to as a "Rightward Opening" or "Horizontal" parabola.

3. Length and Geometry of the Latus Rectum

The Latus Rectum is one of the most critical geometric parameters of a parabola. It provides a definitive measure of the curve's "openness" or "width" at its focus. In coordinate geometry, while any line segment joining two points on the parabola is a chord, the latus rectum is a highly specific focal chord that must satisfy the condition of being perpendicular to the axis of symmetry.

Algebraic Derivation of Length and Coordinates

To find the dimensions of the latus rectum for the standard parabola $y^2 = 4ax$ (where $a > 0$), we utilize the coordinates of the focus and the symmetry of the curve.

Step 1: Establishing the Equation of the Latus Rectum

The focus of the parabola $y^2 = 4ax$ is located at $S(a, 0)$. Since the latus rectum is perpendicular to the axis of symmetry (the x-axis), it must be a vertical line passing through the focus. Therefore, its equation is:

Step 2: Finding the Points of Intersection

The endpoints of the latus rectum, $L$ and $L'$, are the points where the line $x = a$ intersects the parabola. We find these by substituting $x = a$ into the parabola's equation:

Simplifying the right-hand side:

Taking the square root of both sides:

Thus, the coordinates of the endpoints are $L(a, 2a)$ and $L'(a, -2a)$.

Step 3: Calculating the Total Length

Since both points have the same x-coordinate, the length of the latus rectum is simply the vertical distance between the y-coordinates of $L$ and $L'$:

Geometric Significance and Features

The latus rectum is more than just a line; it serves as a geometric anchor for the parabola. The following table summarizes its key geometric relationships:

Feature	Geometric Description	Algebraic Value
Semi-Latus Rectum	Half the length of the latus rectum (distance from focus to curve).	$2a$
Relationship to Vertex	The length is four times the distance from vertex to focus ($OV$).	$4 \times a$
Orientation	Always perpendicular to the axis and parallel to the directrix.	$x = a \parallel x = -a$
Focal Double Ordinate	It is the specific double ordinate that passes through the focus.	$LL'$

4. Focal Distance of a Point

The focal distance of any point on a parabola is defined as the distance of that point from the focus of the parabola.

Analytical Derivation

Consider the standard right-hand parabola $y^2 = 4ax$ (where $a > 0$) with focus $F(a, 0)$. Let $P(x, y)$ be any arbitrary point lying on this curve.

Step 1: Applying the Distance Formula

The distance between point $P(x, y)$ and the focus $F(a, 0)$ is given by the distance formula:

Step 2: Substitution of Parabola Equation

Since point $P$ lies on the parabola, its coordinates satisfy $y^2 = 4ax$. Substituting this value into the above equation:

Step 3: Simplification

The expression under the radical is a perfect square $(x + a)^2$:

Since for a right-hand parabola $x \geq 0$ and $a > 0$, the sum $x + a$ is always positive. Thus:

Geometric Interpretation (PS = PM)

According to the locus definition of a parabola, the distance from the focus ($PF$) is always equal to the perpendicular distance from the directrix ($PM$).

• The equation of the directrix for $y^2 = 4ax$ is $x = -a$ or $x + a = 0$.
• The perpendicular distance of point $P(x, y)$ from the line $x + a = 0$ is calculated as:

This confirms that $PF = PM = x + a$, reinforcing the geometric foundation of the parabola.

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Summary of Key Facts for $y^2 = 4ax$

Feature	Algebraic / Geometric Value
Vertex	$(0, 0)$
Focus	$(a, 0)$
Equation of Directrix	$x + a = 0$
Equation of Axis	$y = 0$
Length of Latus Rectum	$4a$
Focal Distance of $P(x, y)$	$x + a$

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The Four Standard Forms of a Parabola

While the right-hand parabola ($y^2 = 4ax$) is the most frequently discussed, there are four primary orientations a parabola can take when its vertex is fixed at the origin $(0, 0)$ and its axis lies along one of the coordinate axes. These forms are determined by which variable is squared and the sign of the constant coefficient.

1. Right-Hand Parabola ($y^2 = 4ax$)

This parabola opens towards the positive direction of the x-axis. It exists only for non-negative values of $x$ because $y^2$ cannot be negative.

• Focus: $F(a, 0)$
• Directrix: $x = -a$ (or $x + a = 0$)
• Axis of Symmetry: x-axis ($y = 0$)

2. Left-Hand Parabola ($y^2 = -4ax$)

By changing the sign of the coefficient to negative, the parabola reflects across the y-axis. It opens towards the negative direction of the x-axis. Here, $x$ must be $\leq 0$ for $y$ to be real.

Derivation Summary

If the focus is $F(-a, 0)$ and the directrix is $x = a$, then the distance equality $PF = PM$ yields:

$\sqrt{(x + a)^2 + y^2} = |x - a|$

$(x + a)^2 + y^2 = (x - a)^2$

$x^2 + 2ax + a^2 + y^2 = x^2 - 2ax + a^2$

3. Upward Parabola ($x^2 = 4ay$)

When the $x$ variable is squared instead of $y$, the axis of symmetry becomes the y-axis. This parabola opens towards the positive direction of the y-axis (upwards).

• Focus: $F(0, a)$
• Directrix: $y = -a$ (or $y + a = 0$)
• Axis of Symmetry: y-axis ($x = 0$)
• Focal Distance: $y + a$

4. Downward Parabola ($x^2 = -4ay$)

This is the vertical reflection of the upward parabola. It opens towards the negative direction of the y-axis. For $x$ to be a real number, $y$ must be $\leq 0$.

• Focus: $F(0, -a)$
• Directrix: $y = a$ (or $y - a = 0$)
• Axis of Symmetry: y-axis ($x = 0$)
• Endpoints of Latus Rectum: $(\pm 2a, -a)$

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Comparative Summary Table

The following table provides a comprehensive comparison of all four standard forms, which is essential for quick revision for competitive exams like JEE Main.

Property	$y^2 = 4ax$	$y^2 = -4ax$	$x^2 = 4ay$	$x^2 = -4ay$
Opening	Rightward	Leftward	Upward	Downward
Axis	$y = 0$	$y = 0$	$x = 0$	$x = 0$
Focus	$(a, 0)$	$(-a, 0)$	$(0, a)$	$(0, -a)$
Directrix	$x = -a$	$x = a$	$y = -a$	$y = a$
Latus Rectum	$4a$	$4a$	$4a$	$4a$
Parametric Pt.	$(at^2, 2at)$	$(-at^2, 2at)$	$(2at, at^2)$	$(2at, -at^2)$

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Ellipse

To understand the definition of an ellipse more clearly, consider it as a locus of a point moving in a plane such that the sum of its distances from two fixed points remains unchanged. This unique property distinguishes it from other conic sections like the parabola or hyperbola.

Let the two fixed points (foci) be $F_1$ and $F_2$. If we take any arbitrary point $P$ on the curve of the ellipse, the distance $PF_1$ added to the distance $PF_2$ will always result in the same numerical value, regardless of where $P$ is located on the boundary.

Proof that the Constant Sum is Equal to $2a$

Let the distance between the two vertices $A_1$ and $A_2$ be $2a$ (the length of the major axis). Let the center of the ellipse be at the origin $(0, 0)$. Thus, the coordinates of the vertices are $A_1(-a, 0)$ and $A_2(a, 0)$. Let the foci be $F_1(-c, 0)$ and $F_2(c, 0)$.

Since the vertex $A_2(a, 0)$ is a point on the ellipse, it must satisfy the definition that the sum of distances from the foci is constant.

From the coordinate geometry of the points:

Substituting these values into equation (i):

$\text{Constant} = (a + c) + (a - c)$

Therefore, for any point $P(x, y)$ on the ellipse, the governing equation is always:

Practical Visualization: The Pins and String Method

A practical way to draw an ellipse (often called the Gardener's Construction) illustrates this definition perfectly:

1. Fix two pins into a board at points $F_1$ and $F_2$.
2. Take a piece of string longer than the distance between the pins and tie the ends to the pins (or loop a closed string around them).
3. Place a pencil point $P$ against the string and pull it taut so that the string forms a triangle $PF_1F_2$.
4. Move the pencil around the pins while keeping the string tight. The path traced will be an ellipse.

Because the length of the string is fixed, the sum of the lengths of the two sides of the triangle ($PF_1 + PF_2$) is always constant, which is exactly the definition provided.

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Key Terminology and Relationship between Parameters

To analyze the geometry of an ellipse, we must define the specific segments and points that determine its shape and size. These parameters are not independent; they are linked by geometric constraints.

The following table summarizes the essential components of an ellipse as per the standard horizontal form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b$:

Term	Description	Notation / Value
Center (O)	The point of symmetry and midpoint of the foci.	$(0, 0)$
Foci	Two fixed points $F_1$ and $F_2$ that define the locus.	$(\pm c, 0)$
Major Axis	The longest diameter passing through the foci.	Length = $2a$
Minor Axis	The shortest diameter perpendicular to the major axis.	Length = $2b$
Vertices	Endpoints of the major axis.	$(\pm a, 0)$
Semi-axes	Half-lengths of the major and minor axes.	$a$ and $b$
Focal Length	Distance between the center and either focus.	$c$

Derivation of the Relationship between $a, b,$ and $c$

Consider the ellipse with center at the origin and foci at $F_1(-c, 0)$ and $F_2(c, 0)$. Let $B$ be a point at the end of the minor axis, having coordinates $(0, b)$.

According to the definition of an ellipse, the sum of distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis, $2a$.

By the distance formula, the distance from $B(0, b)$ to $F_2(c, 0)$ is:

$BF_2 = \sqrt{(c - 0)^2 + (0 - b)^2} = \sqrt{c^2 + b^2}$

Similarly, the distance from $B(0, b)$ to $F_1(-c, 0)$ is:

$BF_1 = \sqrt{(-c - 0)^2 + (0 - b)^2} = \sqrt{c^2 + b^2}$

Since the ellipse is symmetric, $BF_1 = BF_2$. Substituting these into equation (i):

$\sqrt{c^2 + b^2} + \sqrt{c^2 + b^2} = 2a$

$2\sqrt{c^2 + b^2} = 2a$

$\sqrt{c^2 + b^2} = a$

Squaring both sides, we establish the fundamental relationship:

From this, we can derive the focal distance:

$c = \sqrt{a^2 - b^2}$

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Eccentricity of an Ellipse ($e$)

The eccentricity of an ellipse is a fundamental numerical parameter that describes its "roundness" or "flatness." In simple terms, it tells us how much an ellipse deviates from being a perfect circle.

Geometrically, it is defined as the ratio of the distance from the center to a focus ($c$) to the distance from the center to a vertex ($a$).

Since the focus always lies between the center and the vertex ($c < a$), the value of eccentricity for any ellipse is always less than 1 but greater than or equal to 0.

The eccentricity provides a measure of how "stretched" the ellipse is. If $e$ is close to 0, the ellipse looks like a circle. As $e$ approaches 1, the ellipse becomes increasingly elongated.

Variation of Shape with Eccentricity

The following table illustrates how the numerical value of eccentricity determines the geometric appearance of the conic section:

Value of $e$	Resulting Shape	Description
$e = 0$	Circle	Foci coincide at the center ($c = 0$).
$e \to 0$	Near-Circular Ellipse	The ellipse is very "fat" and round.
$e \to 1$	Elongated Ellipse	The ellipse is very "flat" or thin.
$e = 1$	Line Segment	The ellipse flattens completely into a line.

Expressing $c$ in terms of $e$

From the definition of eccentricity, we can rewrite the focal distance as:

Substituting this into the relationship $b^2 = a^2 - c^2$, we get an alternative formula for $b$ in terms of $e$:

$b^2 = a^2 - (ae)^2$

$b^2 = a^2(1 - e^2)$

Special Cases of an Ellipse: Circle and Line Segment

The shape of an ellipse is determined by the relationship between its semi-major axis ($a$), semi-minor axis ($b$), and the distance of the foci from the center ($c$). By varying these parameters, an ellipse can transform into two limiting geometric figures: a circle and a line segment.

Case I: The Circle (When $c = 0$)

A circle is a special case of an ellipse where the two foci $F_1$ and $F_2$ merge into a single point at the center of the ellipse. This occurs when the distance between the foci ($2c$) becomes zero.

Mathematical Derivation

We know the fundamental relationship between the axes and the focal distance is:

$a^2 = b^2 + c^2$

If the foci coincide at the center, then $c = 0$. Substituting this into the equation:

$a^2 = b^2 + 0^2$

Now, let's look at the Eccentricity ($e$):

Substituting $a = b$ into the standard equation of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$:

$\frac{x^2}{a^2} + \frac{y^2}{a^2} = 1$

Case II: The Line Segment (When $c = a$)

If we move the foci further apart until they reach the vertices of the ellipse, the ellipse flattens out completely. This occurs when the focal distance ($c$) is equal to the semi-major axis ($a$).

Mathematical Derivation

Using the relationship $b^2 = a^2 - c^2$, if $c = a$:

$b^2 = a^2 - a^2$

The Eccentricity ($e$) in this case becomes:

When the semi-minor axis $b$ becomes $0$, the curve no longer has any "width" in the $y$-direction. The ellipse reduces to the line segment $F_1F_2$ along the x-axis, connecting the two vertices.

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Summary Comparison

Figure	Condition	Eccentricity ($e$)	Axis Relationship
Circle	$c = 0$	$0$	$a = b$
Ellipse	$0 < c < a$	$0 < e < 1$	$a > b$
Line Segment	$c = a$	$1$	$b = 0$

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Derivation of the Standard Equation of an Ellipse

To derive the standard equation of an ellipse, we place the center of the ellipse at the origin $(0, 0)$ and the major axis along the x-axis. This is known as the horizontal ellipse.

Assumptions and Setup

Let the two fixed points (foci) be $F_1(-c, 0)$ and $F_2(c, 0)$. Let the total length of the major axis be $2a$. By the definition of an ellipse, for any arbitrary point $P(x, y)$ on the curve, the sum of its distances from the two foci is constant and equal to $2a$.

Mathematically, the definition is expressed as:

Step-by-Step Derivation

Step 1: Applying the Distance Formula

Using the coordinates $P(x, y)$, $F_1(-c, 0)$, and $F_2(c, 0)$, we substitute the distances into equation (i):

$\sqrt{(x + c)^2 + (y - 0)^2} + \sqrt{(x - c)^2 + (y - 0)^2} = 2a$

$\sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2a$

Step 2: Isolating one radical

To simplify, move one of the square root terms to the right side:

$\sqrt{(x + c)^2 + y^2} = 2a - \sqrt{(x - c)^2 + y^2}$

Step 3: First Squaring

Squaring both sides of the equation:

$(x + c)^2 + y^2 = \left( 2a - \sqrt{(x - c)^2 + y^2} \right)^2$

$(x + c)^2 + y^2 = 4a^2 + (x - c)^2 + y^2 - 4a\sqrt{(x - c)^2 + y^2}$

Expanding the squared binomials $(x + c)^2$ and $(x - c)^2$:

$x^2 + 2cx + c^2 + y^2 = 4a^2 + x^2 - 2cx + c^2 + y^2 - 4a\sqrt{(x - c)^2 + y^2}$

Step 4: Simplifying and Isolating the remaining radical

Cancel $x^2, c^2,$ and $y^2$ from both sides:

$2cx = 4a^2 - 2cx - 4a\sqrt{(x - c)^2 + y^2}$

$4cx - 4a^2 = - 4a\sqrt{(x - c)^2 + y^2}$

Divide the entire equation by $-4$:

$a^2 - cx = a\sqrt{(x - c)^2 + y^2}$

Divide by $a$ to fully isolate the radical:

Step 5: Second Squaring

Square both sides of equation (ii) again:

$\left( a - \frac{c}{a}x \right)^2 = (x - c)^2 + y^2$

$a^2 - 2cx + \frac{c^2}{a^2}x^2 = x^2 - 2cx + c^2 + y^2$

Cancel $-2cx$ from both sides and rearrange the terms involving $x$ and $y$ on one side:

$a^2 - c^2 = x^2 - \frac{c^2}{a^2}x^2 + y^2$

$a^2 - c^2 = x^2 \left( 1 - \frac{c^2}{a^2} \right) + y^2$

$a^2 - c^2 = x^2 \left( \frac{a^2 - c^2}{a^2} \right) + y^2$

Step 6: Substituting the Semi-minor Axis ($b$)

We know the relationship between the parameters is $a^2 = b^2 + c^2$, which implies $b^2 = a^2 - c^2$. Substituting $b^2$ into the equation:

$b^2 = x^2 \left( \frac{b^2}{a^2} \right) + y^2$

Step 7: Final Standard Form

Divide the entire equation by $b^2$:

This equation represents the locus of points forming an ellipse with its major axis along the x-axis and center at $(0, 0)$.

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Key Facts and Properties of the Ellipse

For the standard horizontal ellipse represented by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b$), several geometric and algebraic properties can be observed based on its symmetry and structure.

1. Symmetry of the Ellipse

The standard equation of an ellipse is given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The most striking geometric feature of this curve is its symmetry. In coordinate geometry, symmetry implies that for every point $(x, y)$ on the curve, certain related points must also exist on the curve.

Because the variables $x$ and $y$ appear as squares in the equation, their mathematical signs do not change the value of the terms. This leads to three types of symmetry:

(a) Symmetry about the Y-axis (Major/Minor axis)

If we replace $x$ with $-x$ in the equation:

$\frac{(-x)^2}{a^2} + \frac{y^2}{b^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Since the equation remains unchanged, it implies that if a point $(x, y)$ lies on the ellipse, then the point $(-x, y)$ must also lie on the curve. Geometrically, this means the ellipse is a mirror image of itself across the y-axis.

(b) Symmetry about the X-axis (Major/Minor axis)

If we replace $y$ with $-y$ in the equation:

$\frac{x^2}{a^2} + \frac{(-y)^2}{b^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

The equation remains the same, implying that if $(x, y)$ is on the ellipse, then $(x, -y)$ is also on the curve. This shows that the ellipse is perfectly balanced above and below the x-axis.

(c) Symmetry about the Origin (Center)

If we replace both $x$ with $-x$ and $y$ with $-y$ simultaneously:

$\frac{(-x)^2}{a^2} + \frac{(-y)^2}{b^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

This confirms that for every point $(x, y)$, there is a corresponding point $(-x, -y)$. This is known as point symmetry about the origin. Because of this property, the origin $(0, 0)$ is called the Center of the ellipse.

Visual Interpretation

As shown in the diagram above, if you pick any point $P$ in the first quadrant, its reflections across the axes and the center will precisely land on other parts of the ellipse. This quadruple symmetry ensures that:

• The foci are equidistant from the center ($OF_1 = OF_2 = c$).
• The vertices are equidistant from the center ($OA_1 = OA_2 = a$).
• The co-vertices are equidistant from the center ($OB_1 = OB_2 = b$).

This symmetry is fundamental in simplifying calculations for areas, focal distances, and the derivation of properties like the length of the latus rectum.

2. Boundedness and Intercepts of an Ellipse

The ellipse is a closed and bounded curve, which means it does not extend infinitely in any direction. This property is mathematically evident from its standard equation and defines the region of the coordinate plane within which the ellipse exists.

To understand why an ellipse is a "closed" and "bounded" figure, we must analyze its equation under the condition that the coordinates $(x, y)$ are real numbers. In the real number system, the square of any value is always non-negative (i.e., $\geq 0$).

Consider the standard equation of the ellipse:

(a) Determining the Domain (Horizontal Bounds)

The domain refers to all possible values of $x$ for which $y$ is a real number. To find this, we isolate the $x$-term from equation (i):

Since $y^2$ and $b^2$ are squares of real numbers, their ratio $\frac{y^2}{b^2}$ must be greater than or equal to zero. If we subtract a non-negative number from 1, the result must be less than or equal to 1.

Therefore, it follows that:

Multiplying both sides by $a^2$ (which is positive):

When we take the square root of both sides of an inequality like $x^2 \leq a^2$, the value of $x$ is trapped between the negative and positive roots:

(b) Determining the Range (Vertical Bounds)

The range refers to all possible values of $y$ for which $x$ is a real number. Similarly, we isolate the $y$-term from equation (i):

Just as before, because $\frac{x^2}{a^2}$ is always non-negative ($\geq 0$), the right-hand side of the equation cannot exceed 1.

Multiplying by $b^2$:

Solving the inequality for $y$:

(c) Geometric Significance: The Bounding Box

These two derivations prove that the ellipse cannot "escape" a specific rectangular region. The x-coordinates are restricted between $-a$ and $a$, and the y-coordinates are restricted between $-b$ and $b$.

The Bounding Rectangle

If you were to draw four lines: $x = a$, $x = -a$, $y = b$, and $y = -b$, they would form a rectangle. The ellipse is perfectly inscribed within this rectangle, touching it exactly at the four intercepts (vertices and co-vertices).

This property distinguishes the ellipse from other conic sections like the parabola or hyperbola, which have equations that allow at least one variable to increase to infinity ($\infty$).

These inequalities show that the domain of the ellipse is $[-a, a]$ and the range is $[-b, b]$. Consequently, the entire curve is confined within a rectangle of dimensions $2a \times 2b$.

(d) Intercepts on the Axes

The points where the ellipse crosses the coordinate axes are known as the intercepts. These points define the major and minor diameters of the shape.

Type of Intercept	Condition	Coordinates	Terminology
X-Intercepts	Put $y = 0$	$(\pm a, 0)$	Vertices ($A_1, A_2$)
Y-Intercepts	Put $x = 0$	$(0, \pm b)$	Co-vertices ($B_1, B_2$)

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3. Variation of Coordinates in the First Quadrant

To understand the shape and behavior of the ellipse curve, it is useful to examine how the $y$-coordinate changes as the $x$-coordinate varies. For this analysis, we focus on the first quadrant, where both $x$ and $y$ are non-negative ($x \geq 0, y \geq 0$).

Mathematical Functional Relationship

Starting from the standard equation of the horizontal ellipse:

We can express $y$ as a function of $x$ by rearranging the terms:

$\frac{y^2}{b^2} = 1 - \frac{x^2}{a^2}$

$\frac{y^2}{b^2} = \frac{a^2 - x^2}{a^2}$

Taking the square root on both sides, and considering only the positive root for the first quadrant:

Analysis of Monotonicity

From the above equation, we can observe the inverse relationship between the magnitude of $x$ and $y$ within the first quadrant:

• As $x$ increases: The term $x^2$ increases, which makes the value of $(a^2 - x^2)$ smaller. Consequently, $y$ decreases.
• As $x$ decreases: The term $(a^2 - x^2)$ becomes larger, approaching $a^2$. Consequently, $y$ increases.

This confirms that the curve is monotonically decreasing in the first quadrant, sloping downwards from the positive y-axis toward the positive x-axis.

Boundary Values (Intercepts)

We can determine the extreme points of the curve in this quadrant by substituting the boundary values of $x$:

Thus, $b$ is the largest ordinate (vertical distance) which occurs at the center's vertical alignment, and the ordinate becomes zero when the curve reaches the vertex $a$.

Summary Table of Coordinate Variation

Change in $x$	Effect on $y$	Geometric Observation
$0 \to a$ (Increases)	$b \to 0$ (Decreases)	Curve moves from Co-vertex to Vertex
$a \to 0$ (Decreases)	$0 \to b$ (Increases)	Curve moves from Vertex to Co-vertex

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4. Length of the Latus Rectum

The latus rectum of an ellipse is a focal chord that is perpendicular to the major axis. Since an ellipse has two foci, it has two latus recta of equal length. For a standard horizontal ellipse, the latus rectum passes through the foci $(\pm c, 0)$ and its endpoints lie on the curve.

Derivation of the Length

Consider the standard equation of the ellipse where the major axis is along the x-axis:

Let the latus rectum pass through the focus $F_2(c, 0)$. Let $L$ be the endpoint of the latus rectum in the first quadrant. Since $L$ lies directly above the focus, its x-coordinate is $c$. Let its y-coordinate be $k$. Thus, the coordinates of $L$ are $(c, k)$.

Since the point $L(c, k)$ lies on the ellipse, it must satisfy the above equation:

Now, isolate the term containing $k$:

We know from the fundamental relationship of an ellipse that $b^2 = a^2 - c^2$. Substituting this into the equation:

Multiplying both sides by $b^2$:

Taking the positive square root to find the height $k$ (semi-latus rectum):

The total length of the latus rectum (chord $LL'$) is twice the value of $k$ because the ellipse is symmetric about the major axis:

Summary of Results

Whether the ellipse is horizontal or vertical, the formula for the length of the latus rectum depends on the semi-major axis ($a$) and the semi-minor axis ($b$).

Parameter	Formula
Semi-latus Rectum	$\frac{b^2}{a}$
Total Length of Latus Rectum	$\frac{2b^2}{a}$
Coordinates of Endpoints (Horizontal)	$\left( c, \pm \frac{b^2}{a} \right)$ and $\left( -c, \pm \frac{b^2}{a} \right)$

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5. Focal Distances of a Point

The focal distance of any point $P(x, y)$ on an ellipse is the distance of that point from each of the two foci. Since an ellipse has two foci, every point on the curve has two focal distances. A fundamental property of the ellipse is that the sum of these two focal distances is always constant and equal to the length of the major axis ($2a$).

Mathematical Derivation

Consider the standard horizontal ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with foci at $F_1(-c, 0)$ and $F_2(c, 0)$. Let $P(x, y)$ be any point on this ellipse. We know the relationship $c = ae$.

1. Derivation for Focal Distance $PF_1$

Using the distance formula from $P(x, y)$ to $F_1(-c, 0)$:

$PF_1^2 = (x + c)^2 + (y - 0)^2$

$PF_1^2 = (x + c)^2 + y^2$

From the equation of the ellipse, we can express $y^2$ as $b^2\left(1 - \frac{x^2}{a^2}\right)$. Substituting this into the expression:

$PF_1^2 = (x + c)^2 + b^2\left(1 - \frac{x^2}{a^2}\right)$

Substituting $b^2 = a^2 - c^2$:

$PF_1^2 = x^2 + 2xc + c^2 + (a^2 - c^2)\left(\frac{a^2 - x^2}{a^2}\right)$

$PF_1^2 = x^2 + 2xc + c^2 + \frac{a^4 - a^2x^2 - c^2a^2 + c^2x^2}{a^2}$

$PF_1^2 = x^2 + 2xc + c^2 + a^2 - x^2 - c^2 + \frac{c^2x^2}{a^2}$

Simplifying the terms:

$PF_1^2 = a^2 + 2xc + \frac{c^2x^2}{a^2}$

$PF_1^2 = \left(a + \frac{c}{a}x\right)^2$

Since eccentricity $e = \frac{c}{a}$, we get:

2. Derivation for Focal Distance $PF_2$

Using the same logic for the second focus $F_2(c, 0)$:

$PF_2^2 = (x - c)^2 + y^2$

Following the algebraic substitution as shown above, the expression simplifies to:

$PF_2^2 = \left(a - \frac{c}{a}x\right)^2$

Verification of the Constant Sum

If we add the two focal distances obtained in equations (i) and (ii):

$PF_1 + PF_2 = (a + ex) + (a - ex)$

$PF_1 + PF_2 = a + a + ex - ex$

This result confirms that the sum of the focal distances of any point on the ellipse is constant and equal to the length of the major axis.

Summary of Focal Properties

Property	Expression / Value
First Focal Distance ($PF_1$)	$a + ex$
Second Focal Distance ($PF_2$)	$a - ex$
Sum of Focal Distances	$2a$
Range of $x$ for calculations	$-a \leq x \leq a$

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Second Standard Form: Vertical Ellipse

When the foci of an ellipse lie on the y-axis instead of the x-axis, it is referred to as a Vertical Ellipse. In this case, the major axis is along the y-axis, and the minor axis is along the x-axis.

Derivation of the Equation

Let the center of the ellipse be at the origin $(0, 0)$. Let the foci be $F_1(0, -c)$ and $F_2(0, c)$ on the y-axis. Let the length of the major axis be $2a$. For any point $P(x, y)$ on the ellipse, the sum of distances from the foci is constant.

Using the distance formula:

$\sqrt{(x - 0)^2 + (y + c)^2} + \sqrt{(x - 0)^2 + (y - c)^2} = 2a$

$\sqrt{x^2 + (y + c)^2} = 2a - \sqrt{x^2 + (y - c)^2}$

Squaring both sides:

$x^2 + (y + c)^2 = 4a^2 + x^2 + (y - c)^2 - 4a\sqrt{x^2 + (y - c)^2}$

$x^2 + y^2 + 2cy + c^2 = 4a^2 + x^2 + y^2 - 2cy + c^2 - 4a\sqrt{x^2 + (y - c)^2}$

Simplifying the terms by cancelling $x^2, y^2,$ and $c^2$:

$4cy - 4a^2 = -4a\sqrt{x^2 + (y - c)^2}$

$a - \frac{c}{a}y = \sqrt{x^2 + (y - c)^2}$

Squaring both sides again:

$a^2 - 2cy + \frac{c^2}{a^2}y^2 = x^2 + y^2 - 2cy + c^2$

$x^2 + y^2 \left( 1 - \frac{c^2}{a^2} \right) = a^2 - c^2$

$x^2 + y^2 \left( \frac{a^2 - c^2}{a^2} \right) = a^2 - c^2$

Since we know $a^2 - c^2 = b^2$, we substitute it into the equation:

$x^2 + y^2 \left( \frac{b^2}{a^2} \right) = b^2$

Dividing the entire equation by $b^2$:

Identification of the Major Axis

The major axis always contains the foci and the vertices. To determine the orientation of the ellipse from its standard equation:

• If the denominator of the $x^2$ term is larger, the major axis is along the x-axis (Horizontal Ellipse).
• If the denominator of the $y^2$ term is larger, the major axis is along the y-axis (Vertical Ellipse).

Main Facts about the Ellipse

The following table summarizes the properties for both standard forms where $a > b > 0$ and $c = \sqrt{a^2 - b^2}$:

Property	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$
Feature	Horizontal Ellipse ($a > b$)	Vertical Ellipse ($a > b$)
Centre	$(0, 0)$	$(0, 0)$
Major axis lies along	x - axis	y - axis
Length of major axis	$2a$	$2a$
Length of minor axis	$2b$	$2b$
Foci	$(-c, 0), (c, 0)$	$(0, -c), (0, c)$
Vertices	$(-a, 0), (a, 0)$	$(0, -a), (0, a)$
Length of latus rectum	$\frac{2b^2}{a}$	$\frac{2b^2}{a}$
Relation	$c^2 = a^2 - b^2$	$c^2 = a^2 - b^2$

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Answer:

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Answer:

Hyperbola

A hyperbola is the locus of a point moving in a plane such that the absolute difference of its distances from two fixed points (called foci) remains constant. This constant difference is always equal to the length of the transverse axis ($2a$).

Unlike the ellipse, where the sum of distances is constant, the hyperbola is defined by subtraction. Because it is an absolute difference, the hyperbola consists of two separate curves called branches.

Key Terminology

The geometry of a hyperbola is defined by several specific lines and points:

• Foci ($F_1, F_2$): The two fixed points located at a distance $c$ from the centre.
• Centre ($O$): The midpoint of the line segment joining the foci.
• Transverse Axis: The line segment passing through the foci that intersects the hyperbola. Its length is $2a$.
• Conjugate Axis: The line segment perpendicular to the transverse axis passing through the centre. Its length is $2b$.
• Vertices ($A_1, A_2$): The points where the hyperbola intersects the transverse axis.

Proof that the Constant Difference is $2a$

Given: A hyperbola with centre at origin $(0, 0)$, vertices at $A_1(-a, 0)$ and $A_2(a, 0)$, and foci at $F_1(-c, 0)$ and $F_2(c, 0)$.

To Prove: The constant difference of distances from any point to the foci is $2a$.

Proof:

Let us choose a vertex as our point $P$ on the hyperbola. Consider vertex $A_2(a, 0)$. By the definition of a hyperbola, the difference of distances to the foci $F_1$ and $F_2$ must be the required constant.

From the coordinate setup:

Substituting these values into equation (i):

$\text{Constant} = (a + c) - (c - a)$

$\text{Constant} = a + c - c + a$

Thus, for any arbitrary point $P(x, y)$ on the hyperbola, the governing condition is:

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Relationship between $a$, $b$, and $c$

In the study of a hyperbola, the parameters $a$, $b$, and $c$ represent the semi-transverse axis, the semi-conjugate axis, and the focal length (distance from the centre to a focus), respectively. Unlike the ellipse, the geometric orientation of a hyperbola ensures that the foci are always located further from the centre than the vertices.

Geometric Relationship ($c^2 = a^2 + b^2$)

For any hyperbola, the distance of the focus from the centre ($c$) is always greater than the distance of the vertex from the centre ($a$), i.e., $c > a$. We define the parameter $b$ such that it completes a right-angled triangle relationship with $a$ and $c$.

The fundamental identity that connects these three parameters is:

From this identity, we can derive the value of the semi-conjugate axis ($b$):

Note: In an ellipse, the relationship is $a^2 = b^2 + c^2$ because $a > c$. In a hyperbola, the relationship is $c^2 = a^2 + b^2$ because $c > a$.

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Eccentricity ($e$)

The eccentricity of a hyperbola is a numerical value that describes how "flat" or "open" the branches of the hyperbola are. It is defined as the ratio of the focal distance to the semi-transverse axis.

Properties of Eccentricity in Hyperbola

Since the distance to the focus ($c$) is always greater than the distance to the vertex ($a$) for a hyperbola ($c > a$), the following holds true:

• The eccentricity is always greater than 1 ($e > 1$).
• In terms of eccentricity, the coordinates of the foci can be written as $(\pm ae, 0)$ for a horizontal hyperbola.
• As $e$ increases, the branches of the hyperbola become wider or more "open."

Expressing $b$ in terms of $e$

By substituting $c = ae$ into the fundamental identity $b^2 = c^2 - a^2$, we get:

$b^2 = (ae)^2 - a^2$

$b^2 = a^2(e^2 - 1)$

Comparison of Parameters: Ellipse vs. Hyperbola

Property	Ellipse	Hyperbola
Primary Relation	$a^2 = b^2 + c^2$	$c^2 = a^2 + b^2$
Focal distance ($c$)	$c = \sqrt{a^2 - b^2}$	$c = \sqrt{a^2 + b^2}$
Eccentricity ($e$)	$e = \frac{c}{a} < 1$	$e = \frac{c}{a} > 1$
Conjugate/Minor Axis	$b = a\sqrt{1 - e^2}$	$b = a\sqrt{e^2 - 1}$

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Derivation of the Standard Equation of a Hyperbola

To derive the standard equation of a hyperbola, we place the centre at the origin $(0, 0)$ and the foci on the x-axis. This configuration is known as the Horizontal Hyperbola or the First Standard Form.

Geometric Setup

Let the two fixed points (foci) be $F_1(-c, 0)$ and $F_2(c, 0)$. Let $2a$ be the constant difference of the distances from any point $P(x, y)$ on the hyperbola to the two foci. By the geometric definition of a hyperbola:

Mathematical Derivation

Step 1: Applying the Distance Formula

Using the coordinates of $P(x, y)$, $F_1(-c, 0)$, and $F_2(c, 0)$, the above equation becomes:

$\sqrt{(x + c)^2 + y^2} - \sqrt{(x - c)^2 + y^2} = \pm 2a$

Step 2: Isolating one radical

Rearrange the equation to isolate one square root term on the left side:

$\sqrt{(x + c)^2 + y^2} = \pm 2a + \sqrt{(x - c)^2 + y^2}$

Step 3: First Squaring

Squaring both sides of the equation:

$(x + c)^2 + y^2 = 4a^2 + (x - c)^2 + y^2 \pm 4a\sqrt{(x - c)^2 + y^2}$

Expanding the squared terms and simplifying:

$x^2 + 2xc + c^2 + y^2 = 4a^2 + x^2 - 2xc + c^2 + y^2 \pm 4a\sqrt{(x - c)^2 + y^2}$

Cancel common terms $x^2, c^2,$ and $y^2$ from both sides:

$2xc = 4a^2 - 2xc \pm 4a\sqrt{(x - c)^2 + y^2}$

$4xc - 4a^2 = \pm 4a\sqrt{(x - c)^2 + y^2}$

Divide the entire equation by $4$:

Step 4: Second Squaring

Squaring both sides of above equation again to remove the remaining radical:

$(xc - a^2)^2 = a^2 \left[ (x - c)^2 + y^2 \right]$

$x^2c^2 - 2a^2xc + a^4 = a^2 \left( x^2 - 2xc + c^2 + y^2 \right)$

$x^2c^2 - 2a^2xc + a^4 = a^2x^2 - 2a^2xc + a^2c^2 + a^2y^2$

Step 5: Rearranging the Terms

Cancel $-2a^2xc$ from both sides and group the variables:

$x^2c^2 - a^2x^2 - a^2y^2 = a^2c^2 - a^4$

$x^2(c^2 - a^2) - a^2y^2 = a^2(c^2 - a^2)$

Step 6: Substituting the Semi-conjugate Axis ($b$)

In a hyperbola, we know that $c > a$, so $(c^2 - a^2)$ is a positive value. We define $b^2 = c^2 - a^2$. Substituting $b^2$ into the equation:

$x^2b^2 - a^2y^2 = a^2b^2$

Step 7: Final Standard Form

Divide the entire equation by $a^2b^2$:

This equation represents a hyperbola with its transverse axis along the x-axis and its conjugate axis along the y-axis.

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Important Facts about the Hyperbola

The properties and geometric behavior of the hyperbola can be analyzed through its standard equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a, b > 0$). Unlike the ellipse, the hyperbola has distinct characteristics such as being an unbounded curve and having restricted regions in its domain.

1. Symmetry of the Hyperbola

The standard equation of a hyperbola, $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, exhibits multiple forms of symmetry due to the fact that both variables $x$ and $y$ are squared. This algebraic property ensures that the geometric shape of the hyperbola is perfectly balanced across the coordinate axes and the origin.

Let the function of the hyperbola be represented as $f(x, y) = \frac{x^2}{a^2} - \frac{y^2}{b^2} - 1 = 0$. We can test for symmetry by substituting negative values for the variables.

(a) Symmetry about the X-axis (Transverse Axis)

If we replace $y$ with $-y$ in the equation, we get:

$\frac{x^2}{a^2} - \frac{(-y)^2}{b^2} = 1 \implies \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Since the equation remains unchanged, it implies that for every point $(x, y)$ on the hyperbola, the point $(x, -y)$ also lies on the curve. This indicates symmetry about the x-axis.

(b) Symmetry about the Y-axis (Conjugate Axis)

If we replace $x$ with $-x$ in the equation, we get:

$\frac{(-x)^2}{a^2} - \frac{y^2}{b^2} = 1 \implies \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Since the equation remains unchanged, it implies that for every point $(x, y)$ on the curve, the point $(-x, y)$ also lies on the curve. This indicates symmetry about the y-axis.

(c) Symmetry about the Origin (Centre)

If we replace both $x$ with $-x$ and $y$ with $-y$ simultaneously:

$\frac{(-x)^2}{a^2} - \frac{(-y)^2}{b^2} = 1 \implies \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Since the equation remains unchanged, the point $(-x, -y)$ also lies on the curve. This confirms that the hyperbola is symmetrical about the origin $(0, 0)$, which is why the origin is defined as the centre of the hyperbola.

Visualizing Symmetry with Branches

Due to these symmetries, the hyperbola is composed of two identical mirror-image branches. Pick any point $P$ on one branch; its reflections across the axes will result in points on either the same branch or the opposite branch.

As shown in the image above, the quadruple symmetry ensures that:

• The distance from the centre to both foci is equal ($c$).
• The distance from the centre to both vertices is equal ($a$).
• The shape of the curve in the first quadrant is mirrored in all other three quadrants.

2. Intercepts and Vertices of a Hyperbola

The intercepts of a curve are the points where it crosses the coordinate axes. For a hyperbola in standard form, these intercepts define the vertices and the physical span of the curve along its transverse axis, while also proving why the curve does not intersect the conjugate axis.

Derivation of Intercepts

Consider the standard equation of the horizontal hyperbola:

(a) Intercepts on the Transverse Axis (x-axis)

To find the points where the hyperbola intersects the x-axis, we set the y-coordinate to zero ($y = 0$). Substituting this into equation (i):

$\frac{x^2}{a^2} - \frac{0^2}{b^2} = 1$

$\frac{x^2}{a^2} = 1$

These two points, $A_1(-a, 0)$ and $A_2(a, 0)$, are called the vertices of the hyperbola. They represent the points on the curve closest to the centre. The distance between them is the length of the transverse axis ($2a$).

(b) Intercepts on the Conjugate Axis (y-axis)

To find the points where the hyperbola intersects the y-axis, we set the x-coordinate to zero ($x = 0$). Substituting this into equation (i):

$\frac{0^2}{a^2} - \frac{y^2}{b^2} = 1$

Multiplying by $-b^2$:

Since the square of a real number ($y$) cannot be negative, there are no real solutions for $y$ in the above equation. This mathematically proves that the hyperbola never meets the conjugate axis.

Visual Representation of Vertices

As shown in the diagram, the curve is divided into two separate branches that "turn away" from the centre. The only points where the curve touches the coordinate axes are the two vertices on the transverse axis.

Summary of Intercept Facts

Axis	Intercept Points	Nature of Result
Transverse (x-axis)	$(\pm a, 0)$	Real points (Vertices)
Conjugate (y-axis)	None	Imaginary results ($y^2 = -b^2$)

This property is a key differentiator between the ellipse and the hyperbola. In an ellipse, the curve intersects both axes, creating a closed loop. In a hyperbola, the subtraction sign in the equation prevents intersection with the conjugate axis, resulting in two open branches.

3. Domain and Restricted Region

The domain of a function or relation defines the set of all possible $x$-values for which the curve exists in the real plane. For a hyperbola, the domain is not continuous; instead, it is divided into two separate regions, leaving a restricted zone in the middle where no part of the curve can be found.

Mathematical Derivation of the Domain

To determine the possible values of $x$, we analyze the standard equation of a horizontal hyperbola:

We isolate the term containing $x$ as follows:

For any real point $(x, y)$ on the hyperbola, $y^2$ and $b^2$ are squares of real numbers, which means their ratio must be non-negative:

Adding 1 to a non-negative value results in a sum that is at least 1. Therefore:

Multiplying both sides by the positive constant $a^2$ gives:

Taking the square root of both sides leads to the following conclusion for the magnitude of $x$:

This absolute value inequality represents two distinct intervals:

The Restricted Region

The result from the above equation implies that the $x$-coordinates of any point on the hyperbola can never fall between $-a$ and $a$. Consequently:

• There is no part of the curve in the vertical strip defined by $-a < x < a$.
• The lines $x = a$ and $x = -a$ act as the "starting boundaries" for the two branches of the hyperbola.
• This empty space is what causes the hyperbola to be a disconnected curve, unlike the ellipse which is a continuous loop.

Summary of Domain and Range

For the horizontal hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$:

Aspect	Mathematical Interval
Domain ($x$)	$(-\infty, -a] \cup [a, \infty)$
Restricted Region	$(-a, a)$
Range ($y$)	$(-\infty, \infty)$

This domain property ensures that the hyperbola always consists of two separate branches, each extending infinitely away from the centre along the transverse axis.

4. Unbounded Nature and Monotonicity

One of the most defining characteristics of a hyperbola is that it is an unbounded curve. Unlike the circle or the ellipse, which are closed loops, the branches of a hyperbola extend infinitely into the coordinate plane. This behavior is directly linked to the mathematical relationship between the $x$ and $y$ variables.

Mathematical Expression for $y$

To analyze the behavior of the curve in the first quadrant, we express $y$ in terms of $x$ from the standard equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$:

Taking the positive square root for the first quadrant ($x \geq a$ and $y \geq 0$):

Analysis of Monotonicity

In mathematics, a function is monotonic if it either never decreases or never increases. Let's observe the behavior of $y$ as $x$ varies starting from the vertex $x = a$:

• At the Vertex ($x = a$): Substituting $x = a$ into equation (i), we get $y = \frac{b}{a} \sqrt{a^2 - a^2} = 0$. This confirms the vertex $(a, 0)$ is the starting point of the branch in the first quadrant.
• Increasing $x$: As $x$ increases beyond $a$, the value of $(x^2 - a^2)$ becomes larger. Since the square root of a larger number is larger, $y$ also increases.
• Strictly Increasing: Because $y$ always increases as $x$ increases in this quadrant, the curve is said to be strictly increasing or monotonic.

Unbounded Nature

The term unbounded means the curve has no end. As the $x$-coordinate moves towards positive infinity ($x \to \infty$), the $y$-coordinate also moves towards positive infinity ($y \to \infty$).

5. Length of the Latus Rectum of a Hyperbola

The latus rectum of a hyperbola is a chord that passes through one of the foci and is perpendicular to the transverse axis. Since a hyperbola has two foci, it has two latus recta of equal length. For the standard horizontal hyperbola, the latus rectum passes through the foci $(\pm c, 0)$.

Derivation of the Length

Consider the standard equation of the hyperbola where the transverse axis is along the x-axis:

Let the latus rectum pass through the focus $F_2(c, 0)$. Let $L$ be the endpoint of the latus rectum in the first quadrant. Since $L$ lies directly above the focus $F_2$, its x-coordinate must be $c$. Let its y-coordinate (semi-latus rectum) be $k$. Thus, the point $L(c, k)$ lies on the hyperbola.

Step 1: Substitution into the Equation

Since $L(c, k)$ lies on the curve, it satisfies the above equation:

Step 2: Isolating the $k$ term

Rearranging the equation to solve for $k^2$:

Step 3: Using the Hyperbolic Identity

In a hyperbola, we use the fundamental relation $c^2 = a^2 + b^2$, which implies $b^2 = c^2 - a^2$. Substituting this into equation provided above:

Taking the square root for the height $k$ in the first quadrant:

Step 4: Finding Total Length

The latus rectum is the full chord $LL'$. Due to the symmetry of the hyperbola about the transverse axis, the total length is twice the semi-length $k$:

Comparison Table of Latus Rectum

The formula for the length of the latus rectum remains the same for both horizontal and vertical hyperbolas, as it is a ratio of the square of the semi-conjugate axis to the semi-transverse axis.

Hyperbola Type	Equation	Length of Latus Rectum
Horizontal	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$\frac{2b^2}{a}$
Vertical	$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$	$\frac{2b^2}{a}$

Key Points to Remember

• The semi-latus rectum length is $k = \frac{b^2}{a}$.
• The coordinates of the endpoints for the horizontal form are $\left( c, \pm \frac{b^2}{a} \right)$ and $\left( -c, \pm \frac{b^2}{a} \right)$.
• The length of the latus rectum is also related to eccentricity: since $b^2 = a^2(e^2 - 1)$, the length can be written as $2a(e^2 - 1)$.

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Derivation of Hyperbola in the Second Standard Form (Vertical)

The second standard form of a hyperbola occurs when the transverse axis is along the y-axis and the conjugate axis is along the x-axis. In this configuration, the foci and vertices lie on the y-axis, resulting in a vertical hyperbola.

Geometric Setup and Assumptions

Let the centre of the hyperbola be at the origin $(0, 0)$. Let the two foci be $F_1(0, -c)$ and $F_2(0, c)$ on the y-axis. Let the length of the transverse axis be $2a$, implying the vertices are at $A_1(0, -a)$ and $A_2(0, a)$.

By the definition of a hyperbola, for any point $P(x, y)$ on the curve, the absolute difference of its distances from the two foci is constant and equal to $2a$.

Step-by-Step Mathematical Derivation

Step 1: Applying the Distance Formula

Using the coordinates $P(x, y)$, $F_1(0, -c)$, and $F_2(0, c)$, we write the distance terms:

$\sqrt{(x - 0)^2 + (y + c)^2} - \sqrt{(x - 0)^2 + (y - c)^2} = \pm 2a$

$\sqrt{x^2 + (y + c)^2} = \pm 2a + \sqrt{x^2 + (y - c)^2}$

Step 2: First Squaring

Squaring both sides of the equation to eliminate the first radical:

$x^2 + (y + c)^2 = \left( \pm 2a + \sqrt{x^2 + (y - c)^2} \right)^2$

$x^2 + y^2 + 2yc + c^2 = 4a^2 + x^2 + (y - c)^2 \pm 4a\sqrt{x^2 + (y - c)^2}$

$x^2 + y^2 + 2yc + c^2 = 4a^2 + x^2 + y^2 - 2yc + c^2 \pm 4a\sqrt{x^2 + (y - c)^2}$

Step 3: Simplifying and Isolating the Second Radical

Cancel the common terms $x^2, y^2,$ and $c^2$ from both sides:

$2yc = 4a^2 - 2yc \pm 4a\sqrt{x^2 + (y - c)^2}$

$4yc - 4a^2 = \pm 4a\sqrt{x^2 + (y - c)^2}$

Divide the equation by $4$ to simplify:

Step 4: Second Squaring

Squaring both sides of the above equation to remove the remaining radical:

$(yc - a^2)^2 = a^2 \left[ x^2 + (y - c)^2 \right]$

$y^2c^2 - 2a^2yc + a^4 = a^2(x^2 + y^2 - 2yc + c^2)$

$y^2c^2 - 2a^2yc + a^4 = a^2x^2 + a^2y^2 - 2a^2yc + a^2c^2$

Step 5: Rearranging the Terms

Cancel $- 2a^2yc$ from both sides and group the terms with variables $x$ and $y$:

$y^2c^2 - a^2y^2 - a^2x^2 = a^2c^2 - a^4$

$y^2(c^2 - a^2) - a^2x^2 = a^2(c^2 - a^2)$

Step 6: Introducing the Semi-conjugate Axis ($b$)

In a hyperbola, $c > a$, therefore $(c^2 - a^2)$ is a positive constant. We define this constant as $b^2 = c^2 - a^2$. Substituting $b^2$ into the equation:

$y^2b^2 - a^2x^2 = a^2b^2$

Step 7: Final Standard Form

Divide every term by $a^2b^2$ to obtain the final equation:

Identification of the Transverse Axis

In the equation of a hyperbola, the transverse axis is always along the variable that has the positive coefficient.

• If the $x^2$ term is positive, the hyperbola is horizontal.
• If the $y^2$ term is positive, the hyperbola is vertical.

Unlike the ellipse, $a^2$ is not necessarily the larger denominator; it is the denominator of the positive term.

Comparison of Standard Forms

The orientation of the hyperbola is determined by which term in the equation is positive.

Feature	Horizontal ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$)	Vertical ($\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$)
Centre	$(0, 0)$	$(0, 0)$
Transverse Axis along	x - axis	y - axis
Foci	$(\pm c, 0)$	$(0, \pm c)$
Vertices	$(\pm a, 0)$	$(0, \pm a)$
Length of Transverse Axis	$2a$	$2a$
Length of Conjugate Axis	$2b$	$2b$
Latus Rectum Length	$\frac{2b^2}{a}$	$\frac{2b^2}{a}$
Relation	$c^2 = a^2 + b^2$	$c^2 = a^2 + b^2$

Remark: Unlike the ellipse, $a$ is not necessarily larger than $b$. The transverse axis is identified by the denominator of the positive term.

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Parametric Equations of Conics

The parametric equations of a curve express the coordinates of the points on the curve as functions of a variable, called a parameter. For conics, these equations are extremely useful in simplifying complex geometric problems into single-variable algebraic problems.

1. Parametric form of Circle $x^2 + y^2 = r^2$

Introduction to Parametric Representation

The standard equation of a circle with center at the origin $(0, 0)$ and radius $r$ is $x^2 + y^2 = r^2$. In many geometric problems, it is convenient to express the coordinates $(x, y)$ of any point on the circle in terms of a single variable, known as the parameter. For a circle, this parameter is usually the angle $t$ (or $\theta$) which the radius joining the point to the origin makes with the positive direction of the $x$-axis.

Geometric Construction and Derivation

Let $P(x, y)$ be any point on the circle $x^2 + y^2 = r^2$. Let $OP$ be the radius of length $r$ making an angle $t$ with the positive $x$-axis.

From point $P$, drop a perpendicular $PM$ to the $x$-axis. This forms a right-angled triangle $\triangle OMP$, where:

$OM = x$ (Base)

$MP = y$ (Perpendicular)

$OP = r$ (Hypotenuse)

Applying Trigonometric Ratios:

In $\triangle OMP$:

Similarly,

These two equations, $x = r \cos t$ and $y = r \sin t$, are called the parametric equations of the circle. The point $P(r \cos t, r \sin t)$ is often referred to as the point '$t$'.

Verification of the Equations

To ensure these equations satisfy the Cartesian equation $x^2 + y^2 = r^2$, we square and add equations (i) and (ii):

The parameter $t$ usually varies from $0 \le t < 2\pi$ to cover the entire circumference once.

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2. Parametric form of Circle $(x - h)^2 + (y - k)^2 = r^2$

Concept of Translation in Parametric Form

When the center of a circle is not at the origin but at a point $(h, k)$, the circle is said to be translated. The standard Cartesian equation is given by $(x - h)^2 + (y - k)^2 = r^2$. To find its parametric form, we relate it to the central circle $X^2 + Y^2 = r^2$ by shifting the coordinate axes.

Derivation of the Equations

Let the coordinates of any point $P(x, y)$ on the circle be expressed in terms of a parameter $t$. We can substitute the translated coordinates as follows:

From the parametric form of a circle centered at the origin, we know that:

Now, substituting the values of $X$ and $Y$ back into equations (i) and (ii):

And for the $y$-coordinate:

Thus, the general point on the circle is represented as $(h + r \cos t, k + r \sin t)$, where $t \in [0, 2\pi)$.

Geometric Interpretation

In this shifted circle, the parameter $t$ is the angle that the line segment joining the center $(h, k)$ to the point $P(x, y)$ makes with the positive direction of the $x$-axis (or a line parallel to the $x$-axis passing through the center).

The distance between the center $C(h, k)$ and any point $P(h + r \cos t, k + r \sin t)$ is always constant and equal to the radius $r$:

$CP = \sqrt{[(h + r \cos t) - h]^2 + [(k + r \sin t) - k]^2}$

$CP = \sqrt{(r \cos t)^2 + (r \sin t)^2}$

$CP = \sqrt{r^2(\cos^2 t + \sin^2 t)} = r$

Comparison Table

Feature	Central Circle	Translated Circle
Cartesian Equation	$x^2 + y^2 = r^2$	$(x - h)^2 + (y - k)^2 = r^2$
Center	$(0, 0)$	$(h, k)$
Parametric $x$	$r \cos t$	$h + r \cos t$
Parametric $y$	$r \sin t$	$k + r \sin t$

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3. Parametric form of Parabola $y^2 = 4ax$

Introduction to the Parameter '$t$'

For the standard parabola $y^2 = 4ax$, any point on the curve can be expressed in terms of a single variable $t$. Unlike the circle or ellipse where the parameter is usually an angle, in a parabola, the parameter $t$ is a real number ranging from $-\infty$ to $+\infty$.

Derivation of Parametric Equations

The objective is to find expressions for $x$ and $y$ that satisfy the equality of the parabola equation identically.

Standard Form:

To eliminate the square on the left-hand side (LHS), we need the right-hand side (RHS) to be a perfect square. Looking at $4ax$, we already have $4$ (which is $2^2$). To make $ax$ a perfect square involving $a$, we must choose $x$ such that it contains $a$ and another squared term.

Substitution for x:

Let us assume:

Substituting in the Equation:

Now, substitute the value of $x$ from equation (ii) into equation (i):

We can rewrite the RHS as a single square:

Solving for y:

Taking the square root of both sides, we get the expression for $y$ in terms of $t$:

Hence, the pair of equations $x = at^2$ and $y = 2at$ together represent the parabola $y^2 = 4ax$. For any real value of the parameter $t$, the point $(at^2, 2at)$ will satisfy the parabola's equation.

Geometric Significance

Consider the standard parabola $y^2 = 4ax$. Let $S(a, 0)$ be the focus and $x + a = 0$ be the equation of the directrix. Let $P(x, y)$ be any point on the parabola.

Step 1: Using the Definition of a Parabola

By the definition of a parabola, the distance of a point from the focus is equal to its perpendicular distance from the directrix ($PM$).

Step 2: Cartesian Focal Distance

The distance of $P(x, y)$ from the line $x + a = 0$ is given by $|x + a|$. Since for the parabola $y^2 = 4ax$, $x$ is always non-negative ($x \ge 0$) and $a > 0$, we have:

Step 3: Substitution of Parametric Coordinates

We know that the parametric coordinates of any point $P$ on the parabola $y^2 = 4ax$ are $(at^2, 2at)$. Substituting $x = at^2$ into equation (i):

Thus, the focal distance of a point '$t$' on the parabola is $a(1 + t^2)$. This formula is extremely useful because it reduces the distance calculation from two variables ($x, y$) to a single parameter ($t$).

Visual Representation

The following diagram illustrates the relationship between the focus $S$, the point $P(at^2, 2at)$, and the directrix.

Important Properties

For students preparing for engineering entrance exams in India, the following properties derived from the parametric focal distance are crucial:

Focal Chord Property:

If a focal chord joins two points $t_1$ and $t_2$, then:

Length of Focal Chord:

The length of a focal chord with endpoints $t_1$ and $t_2$ is the sum of their focal distances:

Parametric Forms of Other Parabolas

Depending on the orientation of the parabola, the parametric coordinates change. Below is a summary table for different orientations:

Equation of Parabola	Parametric $x$	Parametric $y$	Point $P(t)$
$y^2 = 4ax$	$at^2$	$2at$	$(at^2, 2at)$
$y^2 = -4ax$	$-at^2$	$2at$	$(-at^2, 2at)$
$x^2 = 4ay$	$2at$	$at^2$	$(2at, at^2)$
$x^2 = -4ay$	$2at$	$-at^2$	$(2at, -at^2)$

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4. Parametric form of Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Introduction to the Eccentric Angle

In the parametric representation of an ellipse, the parameter $t$ (often denoted as $\theta$ in textbooks) is of great geometric significance. It is not the angle that the radius vector of the point makes with the $x$-axis; instead, it is called the eccentric angle of the point $P(x, y)$.

Auxiliary Circle and Geometric Definition

To understand the parameter $t$, we define an Auxiliary Circle. A circle described on the major axis of an ellipse as diameter is called its auxiliary circle.

For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the auxiliary circle is:

Geometric Relation:

Let $P$ be a point on the ellipse. Draw a perpendicular $PN$ from $P$ to the major axis ($x$-axis) and produce it to meet the auxiliary circle at point $Q$. Join $OQ$. The angle $\angle QON = t$ is the eccentric angle of point $P$.

Derivation of Parametric Equations

From the geometric construction, point $Q$ lies on the circle $x^2 + y^2 = a^2$. Since $\angle QON = t$, the coordinates of $Q$ are $(a \cos t, a \sin t)$.

Point $P(x, y)$ on the ellipse has the same $x$-coordinate as point $Q$.

To find the $y$-coordinate, we substitute $x$ into the standard equation of the ellipse:

The equations $x = a \cos t$ and $y = b \sin t$ are the parametric equations of the ellipse. The point is represented as $(a \cos t, b \sin t)$ where $0 \le t < 2\pi$.

Key Comparison

It is crucial to distinguish between the eccentric angle ($t$) and the central angle ($\phi$) made by the point $P$ with the origin.

Angle Type	Definition	Mathematical Relation
Eccentric Angle ($t$)	Angle made by the corresponding point on the auxiliary circle.	$x = a \cos t, y = b \sin t$
Central Angle ($\phi$)	Actual angle made by point $P(x,y)$ with the $x$-axis.	$\tan \phi = \frac{y}{x} = \frac{b \sin t}{a \cos t} = \frac{b}{a} \tan t$

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5. Parametric form of Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Introduction to the Parametric Form

For the standard hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the parametric equations allow us to express any point $P(x, y)$ on the curve using a single variable $t$, known as the eccentric angle. This parameterisation is based on the fundamental trigonometric identity relating secant and tangent functions.

Derivation of the Equations

Step 1: The Standard Equation

We begin with the standard equation of a hyperbola centered at the origin $(0, 0)$ with its transverse axis along the $x$-axis:

In this equation, $a$ represents the length of the semi-transverse axis and $b$ represents the length of the semi-conjugate axis.

Step 2: Selection of Trigonometric Identity

To parameterise the equation, we need a trigonometric identity that mirrors the structure (Variable 1)$^2$ - (Variable 2)$^2$ = 1. From fundamental trigonometry, we know the Pythagorean identity:

Here, $t$ is the parameter, traditionally referred to as the eccentric angle.

Step 3: Comparison and Substitution

By comparing the terms of Equation (i) and Equation (ii), we can perform the following substitutions:

For the $x$-coordinate:

Taking the square root of both sides of the above equation:

For the $y$-coordinate:

Taking the square root of both sides of the above equation:

Step 4: Verification of the Parametric Point

To ensure these equations are correct, we substitute $x = a \sec t$ and $y = b \tan t$ back into the original hyperbola equation:

By cancelling out the common factors in the numerators and denominators:

Since the L.H.S. equals the R.H.S., the coordinates $(a \sec t, b \tan t)$ are proven to represent any arbitrary point $P$ on the hyperbola.

Summary of Parametric Form

Any point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ can be expressed in terms of the parameter $t$ as:

$x = a \sec t$

$y = b \tan t$

The set of these equations is collectively known as the Parametric Equations of the Hyperbola, and the point is often written as $P(t) \equiv (a \sec t, b \tan t)$.

Geometric Interpretation of the Eccentric Angle ($t$)

Unlike the circle, the eccentric angle $t$ for a hyperbola is defined using its Auxiliary Circle, which is $x^2 + y^2 = a^2$.

Construction:

1. Draw the auxiliary circle $x^2 + y^2 = a^2$.

2. Let $P(x, y)$ be a point on the hyperbola. Drop a perpendicular $PN$ from $P$ to the transverse axis ($x$-axis).

3. Draw a tangent $NQ$ from point $N$ to the auxiliary circle, where $Q$ is the point of contact.

4. Join $OQ$ (where $O$ is the origin). The angle $\angle QON = t$ is the eccentric angle of point $P$.

Domain of $t$:

Since $\sec t$ and $\tan t$ are not defined for odd multiples of $\frac{\pi}{2}$, the parameter $t$ is restricted as follows:

$t \in [0, 2\pi)$, but $t \neq \frac{\pi}{2}$ and $t \neq \frac{3\pi}{2}$.

Comparison Table: Ellipse vs. Hyperbola

It is vital to remember the sign and trigonometric function differences between the two conics.

Conic Section	Standard Equation	Parametric Point	Identity Used
Ellipse	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	$(a \cos t, b \sin t)$	$\cos^2 t + \sin^2 t = 1$
Hyperbola	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$(a \sec t, b \tan t)$	$\sec^2 t - \tan^2 t = 1$

Conjugate Hyperbola

If the hyperbola is conjugate, i.e., $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$, the parametric equations swap functions to maintain the identity.

Equations for Conjugate Hyperbola:

The point on the conjugate hyperbola is $(a \tan t, b \sec t)$.

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Summary Table of Parametric Conics

Conic Section	Standard Equation	Parametric Equations	Parameter Range
Circle	$x^2 + y^2 = r^2$	$x=r \cos t, y=r \sin t$	$0 \le t < 2\pi$
Parabola	$y^2 = 4ax$	$x=at^2, y=2at$	$t \in \mathbb{R}$
Ellipse	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	$x=a \cos t, y=b \sin t$	$0 \le t < 2\pi$
Hyperbola	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$x=a \sec t, y=b \tan t$	$t \neq \frac{(2n+1)\pi}{2}$

Applications of Conic Sections

Conic sections are not merely abstract mathematical curves; they are fundamental to the physical laws of the universe and are extensively used in engineering, architecture, and space science. From the path of a cricket ball to the orbits of distant planets, these curves describe the motion and structure of the world around us.

1. Applications of the Circle

The circle is the simplest conic section, defined as the locus of a point moving at a constant distance from a fixed center. Its symmetry makes it indispensable in various fields.

Mechanical Engineering and Transport:

The most iconic application is the wheel. Because the radius is constant, the axle remains at a constant height from the ground, ensuring smooth motion. In India, the Ashoka Chakra on the national flag is a symbolic representation of the wheel of law, designed with perfect circular symmetry.

Gears and Pulley Systems:

Circular gears are used in clocks, vehicles, and industrial machinery to transmit power efficiently between rotating shafts.

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2. Applications of the Parabola

The parabola possesses a unique reflective property: any ray originating from the focus and hitting the inner surface of a parabolic reflector is reflected parallel to the axis of the parabola. Conversely, parallel rays hitting the surface are concentrated at the focus.

Reflective Devices:

This property is used in satellite dishes to collect signals at a central receiver (the focus) and in automobile headlights where a bulb is placed at the focus to produce a powerful parallel beam of light.

Projectile Motion:

In physics, the path followed by any object thrown into the air (a projectile) under the influence of gravity is a parabola. For instance, when a cricketer hits a "sixer," the ball follows a parabolic trajectory before landing in the stands.

Architecture and Bridges:

Parabolic arches are used in the construction of bridges (like the Howrah Bridge approach or modern suspension bridges) because they can distribute weight evenly across the structure.

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3. Applications of the Ellipse

An ellipse is defined by two foci. Its primary property is that any signal emitted from one focus will reflect off the elliptical boundary and pass through the other focus.

Astronomy and Planetary Motion:

According to Kepler’s First Law, the orbits of all planets in our solar system are elliptical, with the Sun situated at one of the two foci. Similarly, artificial satellites launched by ISRO often follow elliptical orbits around the Earth.

Whispering Galleries:

In certain buildings, if a person stands at one focus of an elliptical room and whispers, the sound waves reflect off the walls and can be heard clearly by someone standing at the other focus. A famous Indian example is the Gol Gumbaz in Bijapur, which exhibits similar acoustic properties.

Medical Science (Lithotripsy):

Doctors use elliptical reflectors to treat kidney stones. Shock waves are generated at one focus, and the patient is positioned such that the stone is at the second focus. The waves concentrate on the stone and break it without surgery.

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4. Applications of the Hyperbola

The hyperbola is used in advanced navigation and large-scale structural engineering.

Cooling Towers:

Modern thermal and nuclear power plants in India (like those in Kudankulam) use cooling towers with a hyperboloid shape. This design is preferred because it uses less material while providing exceptional structural strength and stability against high winds.

Navigation Systems (LORAN):

Long Range Navigation (LORAN) systems use the constant difference of distances between a ship and two radio stations (the foci) to determine the ship's position along a hyperbolic curve.

Sonic Booms:

When an aircraft travels faster than the speed of sound, it creates a conical shock wave. The intersection of this cone with the ground is a hyperbola, representing the area where the "sonic boom" is heard.

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