Equation of Ellipse | Properties, Examples, and Interactive App

Definition and Equation of an Ellipse

An ellipse is the set of all points \( M(x,y) \) in a plane such that the sum of the distances from \( M \) to two fixed points \( F_1 \) and \( F_2 \), called the foci, is equal to a constant.

\[ \overline{MF_1} + \overline{MF_2} = \sqrt{(x+c)^2+y^2} + \sqrt{(x-c)^2+y^2} = 2a \]

For \( a \ge b \ge 0 \), eliminating the square roots by squaring and simplifying using the relationship \( a^2 = b^2 + c^2 \), we obtain the standard equation of an ellipse centered at the origin:

\( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \)

The width of the ellipse is \( 2a \) and the height is \( 2b \).
Point \( O(0,0) \) is the center of the ellipse.
Points \( V_1(a,0) \) and \( V_2(-a,0) \) are the vertices of the ellipse.
The foci are located at \( F_1(c,0) \) and \( F_2(-c,0) \).

Graph highlighting the definition of an ellipse — Figure 1. Definition and geometry of an ellipse.

Example 1

An ellipse centered at \( (0,0) \) has \( x \)-intercepts at \( (7,0) \) and \( (-7,0) \), and \( y \)-intercepts at \( (0,4) \) and \( (0,-4) \). Find the equation of the ellipse and the foci \( F_1 \) and \( F_2 \).

View Solution

The equation of an ellipse whose center is at the origin is given by:

\[ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \quad (a > 0, \; b > 0) \]

Setting \( y = 0 \) to find the \( x \)-intercepts:

\[ \dfrac{x^2}{a^2} = 1 \implies x = a \text{ and } x = -a \]

Since the \( x \)-intercepts are \( \pm 7 \), we have \( a = 7 \).

Setting \( x = 0 \) to find the \( y \)-intercepts:

\[ \dfrac{y^2}{b^2} = 1 \implies y = b \text{ and } y = -b \]

Since the \( y \)-intercepts are \( \pm 4 \), we have \( b = 4 \).

To find the foci, use parameter \( c \) related by \( a^2 = b^2 + c^2 \):

\[ 7^2 = 4^2 + c^2 \implies 49 = 16 + c^2 \implies c = \sqrt{49 - 16} = \sqrt{33} \]

The coordinates of the foci are:

\[ F_1(\sqrt{33}, 0) \quad \text{and} \quad F_2(-\sqrt{33}, 0) \]

General Equation of an Ellipse

We can generalize and write the equation of an ellipse whose center is shifted to \( O(h,k) \) as follows:

\( \dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1 \)

with foci at \( F_1(c+h, k) \) and \( F_2(-c+h, k) \), and vertices at \( V_1(a+h, k) \) and \( V_2(-a+h, k) \).

Example 2

Find the center, foci, and vertices of the ellipse given by the equation \( (x - 1)^2 + 4(y - 2)^2 = 16 \), then use a graphing calculator to verify your answers.

View Solution

Rewrite the given equation in standard form by dividing all terms by 16:

\[ \dfrac{(x - 1)^2}{16} + \dfrac{4(y - 2)^2}{16} = \dfrac{16}{16} \]

Simplify and write denominators as squares:

\[ \dfrac{(x - 1)^2}{4^2} + \dfrac{(y - 2)^2}{2^2} = 1 \]

Comparing to the general equation, we identify the parameters: \( a = 4, \; b = 2, \; h = 1, \; k = 2 \).

Center: \( O(h,k) = O(1,2) \)
Parameter \( c \): \( c = \sqrt{a^2 - b^2} = \sqrt{16 - 4} = 2\sqrt{3} \)
Foci: \( F_1(2\sqrt{3}+1, 2) \) and \( F_2(-2\sqrt{3}+1, 2) \)
Vertices: \( V_1(5,2) \) and \( V_2(-3,2) \)

Graph of ellipse in example 2 — Figure 2. Graph of Ellipse for Example 2.

Interactive Tutorial on Equation of an Ellipse

Explore the equation of an ellipse and its properties by adjusting parameters \(a, b, h,\) and \(k\).

The exploration uses the standard equation form:

\( \dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1 \)

Default values: \( a = 4, \; b = 2, \; h = 2, \; k = 3 \). Click "Plot Equation" to start.

Hover your mouse cursor over the graph or plotted points to read exact coordinates.

Exercises and References

Exercise: Show by algebraic calculations that the equation \( \dfrac{(x + 2)^2}{5} + 5(y - 3)^2 = 5 \) represents an ellipse. Find its center, foci, and vertices, then use the interactive app above to verify your answers.

If needed, Free graph paper is available for offline practice.