Chapter 11 Conic Sections (Concepts)

Sections of a Cone

The curves that are the focus of this chapter—the circle, ellipse, parabola, and hyperbola—are collectively known as conic sections or simply conics. Their study is a cornerstone of geometry and has profound applications in fields like astronomy (planetary orbits), physics (parabolic reflectors), and engineering (hyperbolic cooling towers).

They are named "conic sections" because each one can be generated by the intersection of a flat, two-dimensional plane and a three-dimensional double-napped cone. The study of these beautiful and important curves dates back to the ancient Greek mathematicians, notably Apollonius of Perga, who first systematically described them.

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Fundamental Definitions

To understand how conic sections are formed, we must first define the three-dimensional object being sliced: a double-napped right circular cone.

The Double-Napped Cone

A cone is a surface generated by a moving line. To construct it, we need a few key elements:

• Axis: A fixed straight line in space which acts as the line of rotational symmetry for the cone.
• Vertex (V): A fixed point on the axis. This will be the point where the two halves of the cone meet.
• Generator (g): A straight line that passes through the vertex. This is the line that will be rotated to generate the cone's surface.
• Semi-vertical Angle ($\beta$): The constant, acute angle between the fixed axis and the generator. This angle determines how "wide" or "narrow" the cone is.

A double-napped right circular cone is the surface formed when we rotate the generator line (g) 360 degrees around the axis, while keeping the vertex and the semi-vertical angle fixed.

• The term "nappes" refers to the two separate parts of the cone that meet at the vertex. One nappe extends infinitely in one direction, and the other extends infinitely in the opposite direction.
• "Right Circular" means the axis is perpendicular to the circular base that would be formed if the cone were cut by a plane perpendicular to the axis.

The Intersecting Plane

A conic section is the curve formed by the intersection of this cone with a plane (a flat, two-dimensional surface). The specific type of curve we get depends entirely on the orientation of this plane relative to the cone.

We define this orientation using a second angle, $\alpha$:

• Angle of the Plane ($\alpha$): This is the angle that the intersecting plane makes with the vertical axis of the cone.

The entire classification of conic sections is based on the comparison between the plane's angle, $\alpha$, and the cone's semi-vertical angle, $\beta$.

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Types of Conic Sections (Non-Degenerate Cases)

When the slicing plane does not pass through the vertex of the cone, we get one of the four famous conic sections. The specific curve is determined by the relationship between $\alpha$ and $\beta$.

1. Circle ($\alpha = 90^\circ$)

When the cutting plane is perpendicular to the axis of the cone, it cuts straight across one nappe. In this case, the angle $\alpha$ is exactly $90^\circ$.

The resulting cross-section is a perfect circle. A circle can be considered a special, highly symmetric case of an ellipse.

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

When the plane is tilted so it is no longer perpendicular to the axis, but its angle $\alpha$ is still greater than the semi-vertical angle $\beta$, it cuts completely through one nappe on a slant.

This creates a closed, oval-shaped curve called an ellipse. All circles are ellipses, but not all ellipses are circles.

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

When the plane is tilted such that it is exactly parallel to a generator line of the cone, its angle $\alpha$ becomes equal to the semi-vertical angle $\beta$.

The plane intersects only one nappe, forming an open curve that extends infinitely and never closes. This U-shaped curve is a parabola. It represents the boundary case between the closed ellipse and the two-branched hyperbola.

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

When the plane is tilted even further, so that it is steeper than the generator (meaning its angle $\alpha$ is less than the semi-vertical angle $\beta$), it is guaranteed to intersect both nappes of the cone.

This creates two separate, open curves that are mirror images of each other. This two-branched curve is a hyperbola.

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Degenerate Conic Sections

A special situation occurs if the slicing plane passes exactly through the vertex of the cone. In this case, the intersection is called a degenerate conic. The term "degenerate" is used in mathematics to refer to a limiting case where a class of objects changes its nature and becomes simpler.

These are not curves in the usual sense but rather simpler geometric figures like points and lines. The type of degenerate conic formed still depends on the relationship between $\alpha$ and $\beta$.

• A Point: This occurs when the plane passes through the vertex with an angle $\alpha > \beta$. The plane is not steep enough to cut into the cone's surface and only touches its very tip. This is a degenerate ellipse or circle.
• A Straight Line: This occurs when the plane passes through the vertex and is tangent to the cone's surface, meaning $\alpha = \beta$. The intersection is a single straight line, which is one of the cone's generators. This is a degenerate parabola.
• A Pair of Intersecting Lines: This occurs when the plane passes through the vertex and is steep enough to cut through both nappes, meaning $\alpha < \beta$. The intersection consists of two straight generator lines that cross at the vertex. This is a degenerate hyperbola.

While the conic sections are elegantly defined by slicing a cone, our study will focus on their properties and equations as they are described within the two-dimensional Cartesian plane.

Circle and its Various Forms

The circle is one of the four non-degenerate conic sections and is arguably the most familiar. It is defined by a simple and elegant geometric property.

Geometrically, a circle is the set (or locus) of all points in a plane that are at a fixed, constant distance from a fixed point. The fixed point is called the center, and the fixed distance is called the radius.

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1. Standard Equation of a Circle (Center-Radius Form)

We can derive the algebraic equation of a circle directly from its definition by using the distance formula. This form is the most intuitive as it directly relates to the geometric definition.

Derivation

Let the center of the circle be the point $C(h, k)$, and let the radius be a constant value $r$, where $r > 0$.

Let $P(x, y)$ be any point on the circumference of the circle. By definition, the distance from the center C to any such point P must be equal to the radius $r$.

Distance(C, P) = $r$

Using the distance formula between two points, we have:

$\sqrt{(x-h)^2 + (y-k)^2} = r$

To eliminate the square root and obtain a simpler polynomial form, we square both sides of the equation.

$(\sqrt{(x-h)^2 + (y-k)^2})^2 = r^2$

This gives us the standard equation of a circle, also known as the center-radius form:

This form is very useful because the coordinates of the center $(h, k)$ and the square of the radius $r^2$ are explicitly visible in the equation.

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2. Central Form (Circle Centered at the Origin)

A common special case of the standard form is when a circle's center is located at the origin of the Cartesian plane, i.e., at the point $(0, 0)$.

Derivation

In this case, the values for the center are $h=0$ and $k=0$. Substituting these into the standard equation gives:

$(x-0)^2 + (y-0)^2 = r^2$

This simplifies to the central form:

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3. General Equation of a Circle

If we expand the standard equation of a circle and rearrange the terms, we arrive at a more general form of the equation.

Derivation

Starting with the standard form, $(x-h)^2 + (y-k)^2 = r^2$:

Expand the squared binomials:

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

Group the terms by degree and move all terms to the left side:

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

This equation is a second-degree equation in $x$ and $y$. To simplify it, we introduce new constants. Let $g = -h$, $f = -k$, and $c = h^2 + k^2 - r^2$. Substituting these gives:

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

The general equation of a circle is:

From this general form, we can find the center and radius by reversing the substitutions:

• Center: Since $g = -h$ and $f = -k$, the center is $(h, k) = (-g, -f)$.
• Radius: We have $c = h^2 + k^2 - r^2$. Solving for $r$, we get $r^2 = h^2 + k^2 - c$. Substituting $h=-g$ and $k=-f$, we get $r^2 = (-g)^2 + (-f)^2 - c = g^2 + f^2 - c$. So, the radius is:

Key Points for the General Equation

For a general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent a circle, it must satisfy two conditions:

1. The coefficients of $x^2$ and $y^2$ must be equal ($a=b$). We usually divide the equation to make them both 1.
2. There must be no $xy$ term (the coefficient $h$ must be 0).

Nature of the Circle from the General Equation

The value of the expression under the square root, $g^2 + f^2 - c$, determines what the equation represents:

• If $g^2 + f^2 - c > 0$, the radius is real and positive, and the equation represents a real circle.
• If $g^2 + f^2 - c = 0$, the radius is zero, and the equation represents a point circle (a single point at the center).
• If $g^2 + f^2 - c < 0$, the radius would be imaginary. The equation represents an imaginary circle, as no real points $(x,y)$ can satisfy it.

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4. Diameter Form of a Circle

This form is used when the coordinates of the endpoints of a diameter are known.

Derivation

Let the endpoints of a diameter be $A(x_1, y_1)$ and $B(x_2, y_2)$.

A fundamental property of a circle is that the angle subtended by a diameter at any point on the circumference is a right angle ($90^\circ$).

Let $P(x, y)$ be any point on the circle. Then the line segment PA must be perpendicular to the line segment PB.

The condition for two lines to be perpendicular is that the product of their slopes is -1.

Slope of PA, $m_1 = \frac{y - y_1}{x - x_1}$

Slope of PB, $m_2 = \frac{y - y_2}{x - x_2}$

Setting the product of the slopes to -1:

$m_1 \times m_2 = -1$

$\left(\frac{y - y_1}{x - x_1}\right) \left(\frac{y - y_2}{x - x_2}\right) = -1$

$(y - y_1)(y - y_2) = -1 \times (x - x_1)(x - x_2)$

Moving all terms to one side gives the diameter form of the equation of a circle:

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Geometrical Condition for the Intersection of a Line and a Circle

When we consider a straight line and a circle together in a plane, there are three distinct possibilities for how they can interact with each other:

1. The line can pass through the circle, cutting it at two distinct points.
2. The line can just touch the circle at a single point.
3. The line can miss the circle entirely.

We can determine which of these scenarios occurs by using a simple geometric relationship involving the circle's center, its radius, and the line.

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Condition Based on Distance from Center to Line

The key to determining the nature of the intersection is to calculate the perpendicular distance from the center of the circle to the line and compare this distance with the radius of the circle.

Let the circle have a center at $C(h, k)$ and a radius of $r$.

Let the line have the general equation $Ax + By + C = 0$.

First, we calculate the perpendicular distance, $d$, from the center $C(h, k)$ to the line using the formula:

Once we have this distance $d$, we compare it to the radius $r$.

1. The Line is a Secant (Two Intersection Points)

If the distance from the center to the line is less than the radius, the line must pass through the interior of the circle. This means it will intersect the circumference at two distinct points. Such a line is called a secant.

Condition: $d < r$

2. The Line is a Tangent (One Intersection Point)

If the distance from the center to the line is exactly equal to the radius, the line just touches the circle at a single point. This point is called the point of tangency, and the line is a tangent.

Condition: $d = r$

3. The Line and Circle Do Not Intersect

If the distance from the center to the line is greater than the radius, the line is too far away from the center to touch the circle. It passes entirely outside the circle, and there are no intersection points.

Condition: $d > r$

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Relative Position of Two Circles

When two circles are placed in the same plane, their relationship can be fully described by comparing the distance between their centers to their respective radii. This geometric analysis allows us to determine, without solving any simultaneous equations, whether the circles are separate from each other, touch at a single point, or intersect at two distinct points. This method provides a powerful visual and analytical tool for understanding the geometry of circles.

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The Key Comparison: Distance vs. Radii

To analyze the position of two circles, we first need to establish three fundamental quantities:

• Let the first circle have center $C_1$ and radius $r_1$.
• Let the second circle have center $C_2$ and radius $r_2$.
• Let $d$ be the distance between their centers, which is calculated using the distance formula: $d = \text{Distance}(C_1, C_2)$.

The entire relationship between the two circles hinges on comparing the distance $d$ with two critical values: the sum of the radii ($r_1 + r_2$) and the absolute difference of the radii ($|r_1 - r_2|$). Let's explore the possible cases that arise from this comparison.

1. One Circle is Completely Outside the Other

This occurs when the two circles are completely separate, with a gap between them. For this to happen, the distance between their centers must be larger than the distance required to just make them touch, which is the sum of their radii.

Condition: $d > r_1 + r_2$

2. Circles Touch Externally

In this case, the two circles touch at a single point, but neither is inside the other. The centers and the point of contact lie on a straight line. The distance between the centers is precisely the length of the first radius plus the length of the second radius.

Condition: $d = r_1 + r_2$

3. Circles Intersect at Two Distinct Points

For the circles to overlap and cross each other, the distance between their centers must be small enough for them to reach each other ($d < r_1 + r_2$), but large enough that one circle is not completely contained within the other ($d > |r_1 - r_2|$). This range of distances allows for two points of intersection.

Condition: $|r_1 - r_2| < d < r_1 + r_2$

4. Circles Touch Internally

This occurs when one circle is inside the other and they touch at a single point. Imagine the larger circle with radius $r_1$ and center $C_1$. For the smaller circle (radius $r_2$) to touch its inner edge, the distance from $C_1$ to $C_2$ ($d$) plus the radius of the smaller circle ($r_2$) must equal the radius of the larger circle ($r_1$). So, $d + r_2 = r_1$, which rearranges to $d = r_1 - r_2$. The absolute value accounts for either circle being the larger one.

Condition: $d = |r_1 - r_2|$

5. One Circle is Completely Inside the Other (Without Touching)

If one circle is inside the other and there is a gap between their circumferences, the distance between their centers must be even smaller than the distance required for them to touch internally. This means the distance $d$ is less than the absolute difference of their radii.

Condition: $d < |r_1 - r_2|$

Special Case: Concentric Circles

If the distance between the centers is zero, the circles share the exact same center. If their radii are different, they form concentric rings. If their radii are the same, they are the same circle.

Condition: $d = 0$

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Symmetry

Symmetry is a fundamental concept in geometry that describes a sense of balance, harmony, or self-similarity in a shape. A figure is symmetric if it remains unchanged after undergoing a certain transformation, such as reflection, rotation, or translation.

In coordinate geometry, we can use the algebraic equation of a curve to precisely test for various types of symmetry. Understanding a curve's symmetry is a powerful tool that helps us visualize and sketch its graph more easily, as it often means we only need to plot points for a portion of the curve and can then infer the rest.

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Tests for Symmetry

For a curve given by an equation, say $F(x, y) = 0$, we can determine if it possesses symmetry with respect to the coordinate axes or the origin by performing simple algebraic tests.

These tests are based on the core idea that if a curve is symmetric with respect to a line or a point, then for every point $(x, y)$ on the curve, its reflection across that line or through that point must also be a point on the curve, thereby satisfying its equation.

1. Symmetry with respect to the x-axis (Horizontal Symmetry)

A curve is symmetric about the x-axis if the part of the curve above the x-axis is a mirror image of the part below it. This means that if a point $(x, y)$ is on the curve, its reflection across the x-axis, the point $(x, -y)$, must also be on the curve.

Algebraic Test: The equation of the curve remains unchanged when $y$ is replaced by $-y$. This is often true for equations where $y$ only appears with even powers (e.g., $y^2, y^4$).

2. Symmetry with respect to the y-axis (Vertical Symmetry)

A curve is symmetric about the y-axis if the part of the curve to the right of the y-axis is a mirror image of the part to the left. This means that if a point $(x, y)$ is on the curve, its reflection across the y-axis, the point $(-x, y)$, must also be on the curve.

Algebraic Test: The equation of the curve remains unchanged when $x$ is replaced by $-x$. This is often true for equations where $x$ only appears with even powers (e.g., $x^2, x^4$).

3. Symmetry with respect to the Origin (Point Symmetry)

A curve is symmetric about the origin if for any point $(x, y)$ on the curve, the point $(-x, -y)$ is also on the curve. Geometrically, this means that if you draw a line segment from $(x, y)$ through the origin, it will meet the curve again at $(-x, -y)$, and the origin will be the midpoint of this segment.

This is equivalent to a 180-degree rotational symmetry about the origin. If you rotate the entire graph 180 degrees around the origin, it will look identical to how it started.

Algebraic Test: The equation of the curve remains unchanged when $x$ is replaced by $-x$ and $y$ is replaced by $-y$ simultaneously.

4. Symmetry with respect to the Line $y=x$

A curve is symmetric about the line $y=x$ if for any point $(x, y)$ on the curve, the point $(y, x)$ is also on the curve. This type of symmetry is of special importance when studying inverse functions, as the graph of a function and its inverse are always reflections of each other across the line $y=x$.

Algebraic Test: The equation of the curve remains unchanged when $x$ and $y$ are interchanged.

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Symmetry of Standard Conic Sections

The standard conic sections, when centered at the origin or with an axis on a coordinate axis, exhibit clear symmetries that can be verified with the algebraic tests:

• Circle ($x^2 + y^2 = r^2$): This is the most symmetric conic. Replacing $x$ with $-x$, $y$ with $-y$, or both, leaves the equation unchanged. Interchanging $x$ and $y$ also leaves it unchanged. It is symmetric with respect to the x-axis, the y-axis, the origin, and the line $y=x$.
• Parabola ($y^2 = 4ax$): Replacing $y$ with $-y$ gives $(-y)^2 = 4ax \implies y^2 = 4ax$. The equation is unchanged. Thus, it is symmetric with respect to the x-axis. The other tests fail. The x-axis is known as the "axis of symmetry" for this parabola.
• Ellipse ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$): Since both $x$ and $y$ are squared, replacing either or both with their negatives will not change the equation. An ellipse centered at the origin is symmetric with respect to the x-axis, the y-axis, and the origin.
• Hyperbola ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$): Similar to the ellipse, both variables are squared. A hyperbola centered at the origin is also symmetric with respect to the x-axis, the y-axis, and the origin.

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Parabola

The parabola is a U-shaped conic section with a unique and precise geometric definition. It is formed when a plane slices through a cone at an angle that is exactly parallel to the cone's generator line ($\alpha = \beta$).

Parabolas are found in many real-world applications, from the path of a projectile under gravity to the design of satellite dishes, car headlights, and suspension bridges, due to their remarkable reflective and structural properties.

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Definition of a Parabola

A parabola is the set (or locus) of all points in a plane that are equidistant from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix.

Let S be the focus and let line $l$ be the directrix. For any point P on the parabola, the defining property is:

Distance(P, S) = Perpendicular Distance(P, $l$)

PS = PM (where M is the foot of the perpendicular from P to the directrix)

Key Terminology

The definition of a parabola gives rise to several important features:

• Focus (S): The fixed point that defines the parabola. It has a special reflective property: all incoming rays parallel to the axis will reflect off the parabola and converge at the focus.
• Directrix: The fixed line that defines the parabola.
• Axis of Symmetry: The line that passes through the focus and is perpendicular to the directrix. The parabola is a mirror image of itself across this axis.
• Vertex (V): The point where the parabola intersects its axis of symmetry. It is the turning point of the curve and lies exactly halfway between the focus and the directrix.
• Focal Distance: The distance from the vertex to the focus (or from the vertex to the directrix). This distance is denoted by the constant $a$ ($a > 0$).
• Latus Rectum: A chord of the parabola that passes through the focus and is perpendicular to the axis of symmetry. Its length is a measure of the "width" of the parabola at its focus, and is always equal to $4a$.

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

To derive the simplest equation for a parabola, we place it on the Cartesian plane in a standard position: let the vertex be at the origin $(0, 0)$ and the axis of symmetry be the x-axis. This means the parabola will open to the right or left.

Let's derive the equation for a parabola opening to the right.

• The vertex is at $V(0, 0)$.
• The distance from the vertex to the focus is $a$. Since the parabola opens to the right, the focus $S$ is at $(a, 0)$.
• The directrix is a vertical line at the same distance $a$ to the left of the vertex. So, the equation of the directrix $l$ is $x = -a$, or $x+a=0$.

Now, let $P(x, y)$ be any arbitrary point on the parabola. According to the definition of a parabola:

Distance(P, S) = Perpendicular Distance(P, $l$)

The distance PS can be found using the distance formula: $PS = \sqrt{(x-a)^2 + (y-0)^2}$.

The perpendicular distance from a point $(x, y)$ to the vertical line $x+a=0$ is simply $|x+a|$. Since the parabola opens right, $x$ will be non-negative, so this is $x+a$.

Setting these distances equal:

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

To remove the square root, we square both sides:

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

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

Cancel the common terms ($x^2$ and $a^2$) from both sides:

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

Isolate $y^2$:

$y^2 = 4ax$

This is the standard equation for a parabola with its vertex at the origin, opening to the right.

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Standard Equations of a Parabola

The equation of a parabola is simplest when its vertex is at the origin $(0, 0)$ and its axis of symmetry coincides with either the x-axis or the y-axis. There are four such standard forms. The constant $a$ (where $a > 0$) always represents the distance from the vertex to the focus.

1. Parabola Opening to the Right ($y^2 = 4ax$)

This is the standard form derived above.

• Vertex: $(0, 0)$
• Focus: $(a, 0)$
• Directrix: $x = -a$
• Axis of Symmetry: x-axis ($y = 0$)
• Length of Latus Rectum: $4a$

2. Parabola Opening to the Left ($y^2 = -4ax$)

This parabola opens towards the negative x-axis. The derivation is similar, but the focus is at $(-a, 0)$ and the directrix is $x=a$.

• Vertex: $(0, 0)$
• Focus: $(-a, 0)$
• Directrix: $x = a$
• Axis of Symmetry: x-axis ($y = 0$)
• Length of Latus Rectum: $4a$

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

Here, the roles of x and y are interchanged. The axis of symmetry is the y-axis, and the parabola opens towards the positive y-axis.

• Vertex: $(0, 0)$
• Focus: $(0, a)$
• Directrix: $y = -a$
• Axis of Symmetry: y-axis ($x = 0$)
• Length of Latus Rectum: $4a$

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

This parabola opens towards the negative y-axis.

• Vertex: $(0, 0)$
• Focus: $(0, -a)$
• Directrix: $y = a$
• Axis of Symmetry: y-axis ($x = 0$)
• Length of Latus Rectum: $4a$

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Form	$y^2 = 4ax$	$y^2 = -4ax$	$x^2 = 4ay$	$x^2 = -4ay$
Opens	Right	Left	Upwards	Downwards
Vertex	(0, 0)	(0, 0)	(0, 0)	(0, 0)
Focus	(a, 0)	(-a, 0)	(0, a)	(0, -a)
Directrix	$x = -a$	$x = a$	$y = -a$	$y = a$
Axis of Symmetry	x-axis ($y=0$)	x-axis ($y=0$)	y-axis ($x=0$)	y-axis ($x=0$)
Length of Latus Rectum	$4a$	$4a$	$4a$	$4a$
Equation of Latus Rectum	$x = a$	$x = -a$	$y = a$	$y = -a$

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Ellipse

The ellipse is a conic section that results in a closed, oval-shaped curve. It is formed when a plane intersects a cone at an angle that is less steep than the cone's generator but not perpendicular to the cone's axis ($\beta < \alpha < 90^\circ$).

The ellipse has a precise geometric definition based on the sum of distances from two fixed points, which leads to its unique properties and applications in fields like astronomy (planetary orbits), architecture (whispering galleries), and engineering.

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Definition of an Ellipse

An ellipse is the set (or locus) of all points in a plane for which the sum of the distances from two fixed points is a constant. The two fixed points are called the foci (plural of focus).

Let the two foci be $S_1$ and $S_2$. For any point P on the ellipse, the defining condition is:

Distance($P, S_1$) + Distance($P, S_2$) = Constant

This constant sum is always greater than the distance between the two foci and is equal to the length of the major axis ($2a$).

Key Terminology and Concepts

• Foci $(S_1, S_2)$: The two fixed points that define the ellipse. The distance between them is $2c$.
• Center: The midpoint of the line segment connecting the foci. The ellipse is symmetric about its center. For standard ellipses, this is the origin $(0, 0)$.
• Major Axis: The longest diameter of the ellipse. It is the line segment that passes through both foci and has its endpoints on the ellipse. Its length is equal to the constant sum of distances, denoted by $2a$.
• Vertices: The endpoints of the major axis. They are the points on the ellipse farthest from the center.
• Minor Axis: The shortest diameter of the ellipse. It is the line segment that passes through the center, is perpendicular to the major axis, and has its endpoints on the ellipse. Its length is denoted by $2b$.
• Semi-major Axis (a) and Semi-minor Axis (b): Half the lengths of the major and minor axes, respectively. So, $a$ is the distance from the center to a vertex, and $b$ is the distance from the center to an endpoint of the minor axis.
• Relationship between a, b, and c: Let $c$ be the distance from the center to each focus. The values $a, b,$ and $c$ form a right-angled triangle, as shown in the diagram. They are related by the formula: $a^2 = b^2 + c^2$. Since 'a' acts as the hypotenuse, it is always the largest of the three values ($a > b$ and $a > c$).
• Eccentricity (e): A dimensionless number that measures how "circular" or "stretched" an ellipse is. 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 ($a$).

  $e = \frac{c}{a}$

  For any ellipse, $0 < e < 1$.
  • If $e$ is close to 0, $c$ is very small compared to $a$, meaning the foci are close to the center, and the ellipse is nearly a circle.
  • If $e$ is close to 1, $c$ is nearly equal to $a$, meaning the foci are close to the vertices, and the ellipse is very elongated and flat.
• Latus Rectum: A chord passing through a focus, perpendicular to the major axis. An ellipse has two latus recta. Its length is given by the formula $\frac{2b^2}{a}$.

Special Case (Circle): If the two foci $S_1$ and $S_2$ coincide at the center, then $c=0$. From the relation $a^2 = b^2 + c^2$, we get $a^2 = b^2$, so $a=b$. The major and minor axes are equal, and the ellipse becomes a circle. The eccentricity $e = c/a = 0/a = 0$. A circle is an ellipse with an eccentricity of 0.

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Derivation of the Standard Equation

Let's derive the equation for an ellipse with its center at the origin $(0, 0)$ and its foci on the x-axis at $S_1(-c, 0)$ and $S_2(c, 0)$. Let the constant sum of distances be $2a$.

Let $P(x, y)$ be any point on the ellipse. By definition: $PS_1 + PS_2 = 2a$.

Using the distance formula:

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

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

Squaring both sides:

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

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

Canceling terms and rearranging: $4cx - 4a^2 = -4a\sqrt{(x-c)^2 + y^2}$

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

Square both sides again: $(cx - a^2)^2 = (-a)^2((x-c)^2 + y^2)$

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

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

Cancel the $-2a^2cx$ term and rearrange: $a^4 - a^2c^2 = a^2x^2 - c^2x^2 + a^2y^2$

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

From the relationship $a^2 = b^2 + c^2$, we know that $b^2 = a^2 - c^2$. Substituting this in:

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

Finally, divide the entire equation by $a^2b^2$:

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

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Standard Equations of an Ellipse

The simplest equations for an ellipse occur when its center is at the origin $(0, 0)$ and its major and minor axes are aligned with the coordinate axes. The value $a$ is always the semi-major axis, and thus $a^2$ is always the larger denominator.

1. Major Axis along the x-axis (Horizontal Ellipse)

This ellipse is wider than it is tall. The larger denominator ($a^2$) is under the $x^2$ term.

• Vertices: $(\pm a, 0)$
• Foci: $(\pm c, 0)$, where $c^2 = a^2 - b^2$

2. Major Axis along the y-axis (Vertical Ellipse)

This ellipse is taller than it is wide. The larger denominator ($a^2$) is under the $y^2$ term.

• Vertices: $(0, \pm a)$
• Foci: $(0, \pm c)$, where $c^2 = a^2 - b^2$

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Hyperbola

The hyperbola is the final of the four non-degenerate conic sections. It is formed when a plane slices through both nappes of a double-napped cone ($\alpha < \beta$), resulting in an open curve with two separate, unbounded branches that are mirror images of each other.

The geometric definition of a hyperbola is based on the *difference* of distances from two fixed points, which contrasts sharply with the ellipse's definition based on the *sum* of distances. This unique property makes hyperbolas essential in various scientific and technological fields.

Applications include navigation (the LORAN system was based on this principle), astronomy (describing the paths of some comets that are flung out of the solar system), and physics (representing shock waves or the paths of certain subatomic particles).

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Definition of a Hyperbola

A hyperbola is the set (or locus) of all points in a plane for which the absolute difference of the distances from two fixed points remains constant. The two fixed points are called the foci.

Let the two fixed points be the foci $S_1$ and $S_2$. A point P is on the hyperbola if the result of subtracting one distance from the other is always the same positive number. The absolute value covers both branches of the hyperbola elegantly:

|Distance($P, S_1$) - Distance($P, S_2$)| = Constant ($2a$)

This constant difference, $2a$, must always be less than the distance between the foci, $2c$. This condition, $a < c$, ensures that the locus forms a two-branched hyperbola.

Key Terminology and Concepts

• Foci $(S_1, S_2)$: The two fixed points that serve as the geometric anchors for defining the hyperbola.
• Center: The midpoint of the line segment connecting the foci. The hyperbola has 180-degree rotational symmetry about its center.
• Transverse Axis: The line segment that passes through the center and the foci, connecting the two vertices. Its name comes from the fact that it "traverses" or crosses between the two branches. Its length is $2a$, which is the constant difference in the hyperbola's definition.
• Vertices: The two points where the hyperbola intersects the transverse axis. These are the "turning points" of each branch and are the closest points on the two branches to each other.
• Conjugate Axis: A line segment of length $2b$ passing through the center and perpendicular to the transverse axis. While the hyperbola does not actually intersect this axis, it is crucial for defining the shape and slope of the asymptotes.
• Relationship between a, b, and c: Let $c$ be the distance from the center to a focus. Unlike the ellipse, the foci are further from the center than the vertices. The relationship forms a right triangle where $c$ is the hypotenuse: $c^2 = a^2 + b^2$. This is a key difference from the ellipse's formula ($a^2 = b^2 + c^2$).
• Eccentricity (e): A measure of how "open" or "flat" the branches of the hyperbola are. It is the ratio $e = \frac{c}{a}$. Since $c > a$ for any hyperbola, its eccentricity is always greater than 1 ($e > 1$). A value of $e$ just slightly above 1 corresponds to sharp, narrow branches, while a large value of $e$ corresponds to very flat, open branches.
• Asymptotes: Two straight lines that pass through the center and serve as "guidelines" for the hyperbola's branches. As the branches extend to infinity, they get closer and closer to these asymptotes but never actually touch them. The asymptotes form the diagonals of a "central rectangle" whose sides have lengths $2a$ and $2b$.
• Latus Rectum: A chord passing through a focus, perpendicular to the transverse axis. Its length, $\frac{2b^2}{a}$, measures the width of a branch at the focus.

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Derivation of the Standard Equation

Let's derive the equation for a hyperbola with its center at the origin $(0, 0)$ and its foci on the x-axis at $S_1(-c, 0)$ and $S_2(c, 0)$. The constant difference is $2a$.

Let $P(x, y)$ be any point on the hyperbola. By definition: $|PS_1 - PS_2| = 2a$.

The derivation proceeds almost identically to that of the ellipse, starting with:

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

After isolating one radical, squaring, simplifying, and squaring again (as done for the ellipse), we arrive at the step:

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

For a hyperbola, we have the relationship $c^2 = a^2 + b^2$, which means $b^2 = c^2 - a^2$. Substituting this in:

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

Finally, divide the entire equation by $a^2b^2$ to get the standard form:

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

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Standard Equations of a Hyperbola

The simplest equations for a hyperbola are obtained when its center is at the origin $(0, 0)$. The key to identifying the orientation is to look for the term with the positive coefficient. The variable in that term indicates the direction of the transverse axis.

1. Transverse Axis along the x-axis (Horizontal Hyperbola)

The branches open to the left and right. The $x^2$ term is positive.

• Vertices: $(\pm a, 0)$
• Foci: $(\pm c, 0)$, where $c^2 = a^2 + b^2$
• Asymptotes: $y = \pm \frac{b}{a}x$

2. Transverse Axis along the y-axis (Vertical Hyperbola)

The branches open upwards and downwards. The $y^2$ term is positive.

• Vertices: $(0, \pm a)$
• Foci: $(0, \pm c)$, where $c^2 = a^2 + b^2$
• Asymptotes: $y = \pm \frac{a}{b}x$

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Parametric Equations of Conics

In our study of curves, we have primarily used Cartesian equations, which establish a direct relationship between the $x$ and $y$ coordinates of points on the curve (e.g., $x^2 + y^2 = r^2$). An alternative and often powerful way to describe a curve is by using parametric equations.

This method defines the path of a point by describing its $x$ and $y$ coordinates as separate functions of a third, independent variable called a parameter. This approach is particularly useful in physics to describe motion, where the parameter is often time, $t$. As time progresses, the parametric equations tell you the exact $(x, y)$ position of a moving object.

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Parametric Representation of a Curve

A curve can be represented by a pair of equations where the $x$ and $y$ coordinates are given as functions of a parameter, let's say $t$:

$x = f(t)$ and $y = g(t)$

For each value of the parameter $t$ within a specified domain, we get a unique point $(x, y)$ on the curve. The entire curve is traced out as $t$ varies through its range of values. The pair of equations is called a parametric representation of the curve.

Converting from Parametric to Cartesian Form

To convert from a parametric representation back to the more familiar Cartesian form, the goal is to eliminate the parameter. This creates a single equation that directly relates $x$ and $y$. The method for elimination depends on the form of the equations:

• Substitution: Solve one of the parametric equations for the parameter ($t$) and substitute the resulting expression into the other equation.
• Using Identities: For trigonometric parametrizations, use fundamental identities like $\cos^2 \theta + \sin^2 \theta = 1$ or $\sec^2 \theta - \tan^2 \theta = 1$.

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Standard Parametric Equations for Conics

Each of the standard conic sections has a conventional parametric representation that is particularly useful and arises naturally from its geometry.

1. Circle: $x^2 + y^2 = r^2$

For a circle centered at the origin, the most natural parameter is the angle $\theta$ that a line segment from the origin to a point on the circle makes with the positive x-axis. This is the standard angle in polar coordinates.

Using basic trigonometry in the right-angled triangle shown in the diagram, the coordinates $(x, y)$ can be expressed as:

The entire circle is traced once as the parameter $\theta$ varies from $0$ to $2\pi$. We can verify this by eliminating the parameter using the identity $\cos^2 \theta + \sin^2 \theta = 1$.

From the equations, $\cos \theta = \frac{x}{r}$ and $\sin \theta = \frac{y}{r}$. Substituting into the identity gives $(\frac{x}{r})^2 + (\frac{y}{r})^2 = 1 \implies \frac{x^2}{r^2} + \frac{y^2}{r^2} = 1 \implies x^2 + y^2 = r^2$.

2. Parabola: $y^2 = 4ax$

For a parabola, a simple algebraic parameter $t$ is used. The most common parametrization is derived to automatically satisfy the Cartesian equation.

Here, the parameter $t$ can be any real number ($-\infty < t < \infty$). Any point of the form $(at^2, 2at)$ is often referred to as 'the point $t$' on the parabola. We can verify this by substituting these expressions into $y^2 = 4ax$:

LHS: $y^2 = (2at)^2 = 4a^2t^2$.

RHS: $4ax = 4a(at^2) = 4a^2t^2$.

Since LHS = RHS, the parametrization is valid.

3. Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Similar to the circle, the ellipse uses an angular parameter $\theta$, often called the eccentric angle. The representation is a scaled version of the circle's parametrization, stretched horizontally by a factor of 'a' and vertically by a factor of 'b'.

The entire ellipse is traced as $\theta$ varies from $0$ to $2\pi$. We verify this using the same identity as the circle. From the equations, $\cos \theta = \frac{x}{a}$ and $\sin \theta = \frac{y}{b}$. Substituting into $\cos^2 \theta + \sin^2 \theta = 1$ gives $(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$, which is the Cartesian equation.

4. Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

The hyperbola's parametrization is based on the Pythagorean identity involving secant and tangent: $\sec^2 \theta - \tan^2 \theta = 1$.

Here, the parameter $\theta$ varies over a range like $0 \le \theta < 2\pi$, excluding the values where the functions are undefined (e.g., $\theta = \frac{\pi}{2}, \frac{3\pi}{2}$). To verify, we have $\sec \theta = \frac{x}{a}$ and $\tan \theta = \frac{y}{b}$. Substituting into the identity gives $(\frac{x}{a})^2 - (\frac{y}{b})^2 = 1$.

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Applications

The study of conic sections—parabolas, ellipses, and hyperbolas—is far more than a theoretical exercise in mathematics. These curves appear all around us in the natural world and are fundamental to the design of countless technologies. Their unique geometric and reflective properties are harnessed in fields ranging from astronomy and physics to engineering and medicine.

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Applications of the Parabola

The most important property of the parabola is its ability to focus parallel rays to a single point, or conversely, to turn rays from a single point into a parallel beam. This is known as its reflective property.

This single property leads to numerous applications:

• Satellite Dishes and Telescopes: A satellite dish has a parabolic shape to collect weak signals from space. Incoming parallel radio waves bounce off the dish and are all directed to a single point—the focus—where a receiver is placed to capture the concentrated signal.
• Headlights and Searchlights: To create a powerful, focused beam of light, a light bulb is placed at the focus of a parabolic mirror. The light rays emitted from the bulb reflect off the mirror and travel forward as a strong, parallel beam.
• Solar Cookers: A parabolic reflector is used to concentrate sunlight. Parallel rays from the sun are reflected to the focus, where a cooking pot is placed. This concentration of energy generates extremely high temperatures.
• Suspension Bridges: The main cables of a suspension bridge, which support the flat roadway, naturally hang in the shape of a parabola. This shape efficiently distributes the weight of the bridge deck.
• Projectile Motion: In physics, when an object is thrown or launched into the air (neglecting air resistance), its path through space is a parabola.

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Applications of the Ellipse

The ellipse is defined by the property that the sum of the distances from any point on the curve to two fixed foci is constant. This leads to its own unique reflective property and other important applications.

• Planetary Orbits: In one of the most profound discoveries in science, Johannes Kepler found that planets do not orbit the Sun in perfect circles, but in ellipses. The Sun is located at one of the two foci of the elliptical orbit. This is true for planets, comets, and asteroids orbiting a star.
• Whispering Galleries: An elliptical room has a remarkable acoustic property. If a person stands at one focus and whispers, the sound waves will reflect off the elliptical walls and converge at the other focus, allowing another person standing there to hear the whisper clearly, even from a great distance. A famous example is the Gol Gumbaz in Bijapur, India.
• Lithotripsy (Medical Technology): A medical device called a lithotripter uses an elliptical reflector to break up kidney stones without surgery. High-energy shock waves are generated at one focus, and the patient is positioned so the kidney stone is at the other focus. The waves travel from the source, reflect off the elliptical surface, and converge with powerful, concentrated energy on the stone, shattering it.

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Applications of the Hyperbola

The hyperbola, defined by the constant *difference* of distances to two foci, has applications that often involve locating positions or describing certain physical phenomena.

• Navigation Systems (LORAN): Long-range navigation systems use hyperbolas to determine a ship's or aircraft's position. Two radio transmitters at known locations (the foci) send out synchronized signals. A receiver on the ship measures the tiny difference in the arrival time of these signals. This time difference corresponds to a constant difference in distance, placing the ship somewhere on a specific hyperbola. Using a second pair of transmitters, the ship's location can be found at the intersection of two hyperbolas.
• Cooling Towers: The distinctive shape of large cooling towers at power plants is a hyperboloid (a 3D shape created by rotating a hyperbola). This shape provides exceptional structural strength and stability while promoting efficient cooling through natural air convection.
• Astronomy: While many comets have elliptical orbits and return periodically, some comets travel on a hyperbolic path. They pass by the Sun once and have enough velocity to escape its gravitational pull, never to return.

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Applications of the Circle

As a special case of the ellipse (with an eccentricity of zero), the circle is the most common and fundamental shape, with applications that are almost too numerous to list.

• Mechanical Engineering: The circle is the basis for all rotating objects, including wheels, gears, axles, turbines, and bearings.
• Physics: It describes uniform circular motion, wave propagation (ripples in a pond), and is used in the modeling of orbits as a first approximation.
• Everyday Life: Circles are found in the design of clocks, coins, plates, buttons, and countless other objects due to their perfect symmetry and simplicity.

The study of conic sections provides a powerful mathematical language to describe and utilize these fundamental shapes that are woven into the fabric of our universe and technology.