Ellipse in Coordinate Geometry

Ellipse: Definition (Sum of Distances from Two Foci)

An ellipse is a conic section that results from the intersection of a cone with a plane tilted at an angle greater than the semi-vertical angle ($\alpha < \beta < 90^\circ$). Besides this geometric definition via cone sections, an ellipse can be defined as a locus of points based on a property involving distances to two fixed points. This definition involving foci is fundamental to deriving the standard equations and understanding the characteristics of an ellipse.

Definition (Based on Foci)

An ellipse is defined as the locus of all points $P$ in a plane such that the sum of the distances from $P$ to two distinct fixed points, $F_1$ and $F_2$, in the plane is a constant value. The two distinct fixed points $F_1$ and $F_2$ are called the foci (plural of focus) of the ellipse.

Let $P(x, y)$ be any point on the ellipse.

Let $F_1$ and $F_2$ be the two fixed foci.

The defining condition for any point $P$ to be on the ellipse is:

$\mathbf{|PF_1| + |PF_2| = \text{Constant}}$

Let the constant sum be $2a$. (The reason for using $2a$ will become clear when we look at the standard equation). The condition becomes $|PF_1| + |PF_2| = 2a$.

For an ellipse to exist (not degenerate to a line segment), the constant sum ($2a$) must be strictly greater than the distance between the two foci ($|F_1 F_2|$). If $2a = |F_1 F_2|$, the locus is the line segment joining $F_1$ and $F_2$. If $2a < |F_1 F_2|$, there is no locus (empty set).

---

Visualizing the Definition (String Method)

The definition provides a simple method for drawing an ellipse: the "string method".

Place two pins at the locations of the foci ($F_1$ and $F_2$) on a piece of paper.

Take a loop of string whose total length is equal to the desired constant sum ($2a$) plus the distance between the foci ($|F_1 F_2|$). Knot the ends of the string together to form a loop.

Place the loop of string around the two pins.

Insert a pencil (at point P) into the loop and move it while keeping the string taut. The pencil will trace out an ellipse. At any point P, the sum of the distances from P to $F_1$ and P to $F_2$ is equal to the length of the string loop minus the distance between the pins ($|PF_1| + |PF_2| = \text{Loop Length} - |F_1 F_2|$). If the loop length is fixed, this sum is constant, satisfying the definition.

---

Key Terminology Associated with Foci Definition

Based on the foci definition, we can identify and define several key geometric features of an ellipse:

• Foci ($F_1, F_2$): The two fixed points used in the definition ($|PF_1| + |PF_2| = \text{constant}$).

• Center (C): The midpoint of the line segment joining the two foci ($F_1 F_2$). The ellipse is symmetric about its center.

• Major Axis: The line segment that passes through the two foci and has its endpoints on the ellipse. It is the longest diameter of the ellipse. Its length is equal to the constant sum of distances, $2a$.

• Minor Axis: The line segment that passes through the center of the ellipse, is perpendicular to the major axis, and has its endpoints on the ellipse. It is the shortest diameter of the ellipse.

• Vertices: The two endpoints of the major axis. If the length of the major axis is $2a$, the vertices are located at a distance $a$ from the center along the major axis.

• Co-vertices: The two endpoints of the minor axis. If the length of the minor axis is $2b$, the co-vertices are located at a distance $b$ from the center along the minor axis.

The relationship between the constant sum ($2a$), the distance between the foci ($2c$), and the length of the minor axis ($2b$) is $a^2 = b^2 + c^2$. The eccentricity $e = c/a$, and for an ellipse $0 < e < 1$, which implies $0 < c < a$, consistent with $a^2 = b^2 + c^2$ for a real ellipse where $b>0$.

---

Standard Equations of Ellipse ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, etc.)

The equation of an ellipse takes its simplest and most standard form when the ellipse is positioned symmetrically with respect to the coordinate axes, specifically when its center is at the origin $(0, 0)$ and its foci lie on either the x-axis or the y-axis.

Case 1: Foci on the x-axis (Horizontal Major Axis)

Let the center of the ellipse be at the origin $C(0, 0)$.

Let the foci be located on the x-axis, symmetric with respect to the origin. Let their coordinates be $F_1(-c, 0)$ and $F_2(c, 0)$, where $c$ is a positive constant representing the distance from the center to each focus ($c > 0$). The distance between the foci is $2c$.

By the definition of an ellipse, for any point $P(x, y)$ on the ellipse, the sum of the distances from $P$ to the foci is a constant value, which we denote as $2a$. So, $|PF_1| + |PF_2| = 2a$. For an ellipse to be formed (not a degenerate case), the constant sum $2a$ must be strictly greater than the distance between the foci $2c$. Thus, $2a > 2c$, which implies $a > c > 0$.

Using the distance formula for $|PF_1|$ and $|PF_2|$:

$|PF_1| = \sqrt{(x - (-c))^2 + (y - 0)^2} = \sqrt{(x + c)^2 + y^2}$

$|PF_2| = \sqrt{(x - c)^2 + (y - 0)^2} = \sqrt{(x - c)^2 + y^2}$

Substitute these into the ellipse definition equation $|PF_1| + |PF_2| = 2a$:

Derivation Steps (Algebraic Manipulation):

To eliminate the square roots in equation (i), we follow these steps:

1. Isolate one radical term:

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

   ... (ii)

2. Square both sides of equation (ii):

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

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

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

   ... (iii)

3. Simplify equation (iii) by cancelling $x^2, c^2, y^2$ from both sides and rearranging to isolate the remaining radical term:

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

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

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

4. Divide the equation by 4:

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

   ... (iv)

5. Square both sides of equation (iv) again:

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

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

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

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

   ... (v)

6. Simplify equation (v) by cancelling the term $-2a^2cx$ from both sides:

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

   ... (vi)

7. Rearrange the terms in equation (vi) to group terms involving $x$ and $y$ on one side and constant terms on the other:

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

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

   ... (vii)

8. Introduce a new constant $b$. We have $a > c > 0$, so $a^2 > c^2$, which means $a^2 - c^2 > 0$. Let $b^2 = a^2 - c^2$. Since $a^2 - c^2$ is positive, $b^2$ is positive, and $b$ is a real number.

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

   (Substitute $a^2 - c^2 = b^2$ in (vii))

9. Divide the entire equation by $a^2 b^2$. Since $a>c>0$, $a \neq 0$. Also $b^2 = a^2 - c^2 > 0$, so $b \neq 0$. Thus, $a^2 b^2 \neq 0$.

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

   ... (viii)

   Simplify equation (viii):

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

   (Standard form 1)

This is the standard equation of an ellipse centered at the origin with foci on the x-axis. In this case, the major axis lies along the x-axis. Its length is $2a$, and the vertices are $(\pm a, 0)$. The minor axis lies along the y-axis. Its length is $2b$, and the co-vertices are $(0, \pm b)$. Since $b^2 = a^2 - c^2$ and $c>0$, we have $b^2 < a^2$, which means $b < a$. So, the semi-major axis length is $a$ and the semi-minor axis length is $b$. The foci are at $(\pm c, 0) = (\pm \sqrt{a^2 - b^2}, 0)$.

---

Case 2: Foci on the y-axis (Vertical Major Axis)

If the foci are located on the y-axis, symmetric with respect to the origin, at $F_1(0, -c)$ and $F_2(0, c)$, and the constant sum of distances is $2a$ ($a > c > 0$), a similar derivation process leads to a slightly different standard equation.

The equation $|PF_1| + |PF_2| = 2a$ becomes $\sqrt{(x - 0)^2 + (y - (-c))^2} + \sqrt{(x - 0)^2 + (y - c)^2} = 2a$, or $\sqrt{x^2 + (y + c)^2} + \sqrt{x^2 + (y - c)^2} = 2a$.

Performing analogous algebraic manipulations, we arrive at the standard equation for an ellipse centered at the origin with foci on the y-axis:

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

In this equation, $a$ is still the semi-major axis length, $b$ is the semi-minor axis length, and $c$ is the distance from the center to each focus, with the relationship $c^2 = a^2 - b^2$, meaning $a > b > 0$. The major axis now lies along the y-axis. Its length is $2a$, and the vertices are $(0, \pm a)$. The minor axis lies along the x-axis. Its length is $2b$, and the co-vertices are $(\pm b, 0)$. The foci are at $(0, \pm c) = (0, \pm \sqrt{a^2 - b^2})$.

Key Distinction Between the Two Standard Forms:

For standard ellipses centered at the origin, the equations $\frac{x^2}{\text{denominator}_1} + \frac{y^2}{\text{denominator}_2} = 1$ always have the property that the larger denominator corresponds to the square of the semi-major axis length ($a^2$). The smaller denominator corresponds to the square of the semi-minor axis length ($b^2$).

• If the larger denominator ($a^2$) is under $x^2$, the major axis is horizontal (along the x-axis).
• If the larger denominator ($a^2$) is under $y^2$, the major axis is vertical (along the y-axis).

The foci always lie on the major axis, and the vertices are the endpoints of the major axis. The relationship $c^2 = a^2 - b^2$ is used to find the distance from the center to the foci, where $a^2$ is always the larger denominator.

---

Answer:

---

Properties of Ellipse (Foci, Vertices, Axes, Eccentricity, Directrices, Latus Rectum)

Once the standard equation of an ellipse is determined, its key geometric properties can be readily identified. These properties include the location of the foci, vertices, and axes, as well as numerical characteristics like eccentricity and the length of the latus rectum. We focus on ellipses centered at the origin for simplicity; properties for ellipses with a general center $(h, k)$ are obtained by translation.

The standard equations for an ellipse centered at the origin are $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ or $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$. In both cases, $a$ and $b$ are the lengths of the semi-major and semi-minor axes. By convention, $a$ is always taken as the length of the semi-major axis, and $b$ as the length of the semi-minor axis, so $a > b > 0$. The distance from the center to each focus is $c$, related by $c^2 = a^2 - b^2$, with $c > 0$.

Eccentricity ($e$)

The eccentricity of an ellipse is a measure of how "flattened" or "elongated" it is, compared to a perfect circle. It is defined as the ratio of the distance from the center to a focus ($c$) to the length of the semi-major axis ($a$).

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

Since $c^2 = a^2 - b^2$ and $a>b>0$, we have $c = \sqrt{a^2 - b^2}$.

Substituting this into the formula for $e$:

$\mathbf{e = \frac{\sqrt{a^2 - b^2}}{a} = \sqrt{1 - \frac{b^2}{a^2}}}$

For an ellipse, $c > 0$ and $a > c$, which implies $0 < c/a < 1$. Therefore, the eccentricity of an ellipse always satisfies $0 < e < 1$.

• If $e$ is close to 0 (i.e., $c$ is small compared to $a$, or $b$ is close to $a$), the ellipse is nearly circular. A circle itself has eccentricity $e=0$.
• If $e$ is close to 1 (i.e., $c$ is close to $a$, or $b$ is small compared to $a$), the ellipse is very elongated.

The relationship $b^2 = a^2 - c^2$ can be rewritten in terms of eccentricity using $c = ae$:

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

or $\mathbf{a^2 - b^2 = a^2 e^2}$.

---

Directrices

As mentioned in the focus-directrix definition of conic sections, every conic section (except a circle in the strict sense) has at least one focus and one corresponding directrix. An ellipse, having two foci, also has two directrices, one for each focus.

The directrices are lines perpendicular to the major axis and are symmetric with respect to the center.

• For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a>b$, major axis on x-axis, foci at $(\pm c, 0)$):

  The equations of the two directrices are $\mathbf{x = \pm \frac{a}{e}}$. The directrix $x = a/e$ corresponds to the focus $(c, 0) = (ae, 0)$, and the directrix $x = -a/e$ corresponds to the focus $(-c, 0) = (-ae, 0)$. Since $e < 1$, $a/e > a$, confirming that the directrices are vertical lines located outside the ellipse.

• For the ellipse $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ ($a>b$, major axis on y-axis, foci at $(0, \pm c)$):

  The equations of the two directrices are $\mathbf{y = \pm \frac{a}{e}}$. The directrix $y = a/e$ corresponds to the focus $(0, c) = (0, ae)$, and the directrix $y = -a/e$ corresponds to the focus $(0, -c) = (0, -ae)$. These are horizontal lines located outside the ellipse.

---

Latus Rectum

The latus rectum of an ellipse is a chord of the ellipse that passes through one of the foci and is perpendicular to the major axis. Since an ellipse has two foci, it has two latera recta (plural), one for each focus. The length of the latus rectum is the same for both.

Derivation of the Length of the Latus Rectum:

Let's consider the standard ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b$), centered at the origin with foci on the x-axis at $(\pm c, 0)$. The major axis is along the x-axis.

Consider the latus rectum passing through the focus $F_2(c, 0)$. Since it is perpendicular to the major axis (x-axis), it is a vertical line segment with the equation $x = c$. The endpoints of this latus rectum lie on the ellipse.

To find the y-coordinates of the endpoints, substitute $x = c$ into the equation of the ellipse:

Solve equation (ii) for $y^2$:

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

From the relationship $b^2 = a^2 - c^2$, we can substitute $a^2 - c^2$ with $b^2$:

Solve equation (iii) for $y^2$:

Take the square root of equation (iv) to find the y-coordinates:

The endpoints of the latus rectum through $F_2(c, 0)$ are $(c, b^2/a)$ and $(c, -b^2/a)$.

The length of the latus rectum is the distance between these two endpoints. Since they have the same x-coordinate, the distance is the absolute difference of their y-coordinates:

Since $a > 0$ and $b > 0$, $\frac{2b^2}{a}$ is positive.

This length is the same for the other latus rectum (through $F_1$) and also for the standard ellipse with vertical major axis $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ ($a>b$).

---

Summary of Properties (Ellipse Centered at Origin)

For standard ellipses centered at the origin $(0, 0)$, with semi-major axis $a$ and semi-minor axis $b$ (where $a > b > 0$). The distance from the center to the focus is $c = \sqrt{a^2 - b^2}$. Eccentricity $e = c/a$.

Property	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a>b$)	$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ ($a>b$)
(Common: Center $(0,0)$, $c=\sqrt{a^2-b^2}$, $e=c/a$)	(Horizontal Major Axis)	(Vertical Major Axis)
Center	$(0, 0)$	$(0, 0)$
Major Axis Location	Along x-axis	Along y-axis
Length of Major Axis	$2a$	$2a$
Length of Semi-major Axis	$a$	$a$
Length of Minor Axis	$2b$	$2b$
Length of Semi-minor Axis	$b$	$b$
Vertices	$(\pm a, 0)$	$(0, \pm a)$
Co-vertices	$(0, \pm b)$	$(\pm b, 0)$
Foci	$(\pm c, 0) = (\pm ae, 0)$	$(0, \pm c) = (0, \pm ae)$
Eccentricity ($e = c/a$)	$e = \frac{\sqrt{a^2-b^2}}{a}$	$e = \frac{\sqrt{a^2-b^2}}{a}$
Equations of Directrices	$x = \pm \frac{a}{e}$	$y = \pm \frac{a}{e}$
Length of Latus Rectum	$\frac{2b^2}{a}$	$\frac{2b^2}{a}$

---

Answer:

---

Parametric Equations of Ellipse

Representing the coordinates of points on an ellipse using a parameter provides an alternative way to describe the curve. These parametric equations define $x$ and $y$ coordinates as functions of a single independent variable, typically an angle. This form is especially useful in various applications in physics, engineering, and advanced geometry problems.

Parametric Form for $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Consider the standard equation of an ellipse centered at the origin with the major axis along the x-axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. We want to find expressions for $x$ and $y$ in terms of a parameter $t$ (or $\theta$) such that $\left(\frac{x(t)}{a}\right)^2 + \left(\frac{y(t)}{b}\right)^2 = 1$ for all values of $t$.

This form strongly resembles the fundamental trigonometric identity $\cos^2 \theta + \sin^2 \theta = 1$. We can make a direct substitution by letting $\frac{x}{a} = \cos \theta$ and $\frac{y}{b} = \sin \theta$.

Let $\frac{x}{a} = \cos \theta$. This implies $x = a \cos \theta$.

Substitute $x = a \cos \theta$ into the ellipse equation:

Simplify equation (i):

From equation (iii), isolate the term with $y^2$:

Using the Pythagorean trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$, we know that $1 - \cos^2 \theta = \sin^2 \theta$. Substitute this into equation (iv):

Solve equation (v) for $y^2$:

Take the square root of equation (vi):

By choosing the positive sign for simplicity, we get $y = b \sin \theta$. (Choosing the negative sign $y = -b \sin \theta$ also works and traces the same ellipse). The parametric equations for the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are:

$\mathbf{x = a \cos \theta}$

$\mathbf{y = b \sin \theta}$

where $\theta$ is the parameter. As $\theta$ varies, typically from $0$ to $2\pi$ (or any interval of length $2\pi$), the point $(a \cos \theta, b \sin \theta)$ traces the entire ellipse.

The angle $\theta$ used here is sometimes called the eccentric angle of the point P. Geometrically, if you draw a vertical line from P$(a \cos \theta, b \sin \theta)$ up to the auxiliary circle ($x^2 + y^2 = a^2$), the point on the auxiliary circle has coordinates $(a \cos \theta, a \sin \theta)$. The angle $\theta$ is the angle made by the radius to this point on the auxiliary circle with the positive x-axis.

---

Parametric Form for $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$

For the standard ellipse centered at the origin with the major axis along the y-axis, $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (where $a>b$), we use a similar trigonometric substitution. This time we set $\frac{x}{b} = \cos \theta$ and $\frac{y}{a} = \sin \theta$.

Substituting these into the equation $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ gives $(\cos \theta)^2 + (\sin \theta)^2 = 1$, which is always true.

The parametric equations are:

$\mathbf{x = b \cos \theta}$

$\mathbf{y = a \sin \theta}$

Here, the coefficient of $\cos \theta$ is the semi-axis length along the x-axis ($b$), and the coefficient of $\sin \theta$ is the semi-axis length along the y-axis ($a$). The parameter $\theta$ varies from $0$ to $2\pi$.

Parametric Form for General Ellipse (Center $(h, k)$):

If the center of the ellipse is at $(h, k)$, the parametric equations are obtained by translating the origin-centered parametric equations. For example, for the ellipse $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ with a horizontal major axis, the parametric equations are:

$\frac{x-h}{a} = \cos \theta \implies \mathbf{x = h + a \cos \theta}$

$\frac{y-k}{b} = \sin \theta \implies \mathbf{y = k + b \sin \theta}$

For the ellipse $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ with a vertical major axis, the parametric equations are:

$\frac{x-h}{b} = \cos \theta \implies \mathbf{x = h + b \cos \theta}$

$\frac{y-k}{a} = \sin \theta \implies \mathbf{y = k + a \sin \theta}$

---

Answer:

---