# Find the Points of Intersection of Two Ellipses

This is tutorial on finding the points of intersection of two ellipses given by their equations.

## Example 1

Find the points of intersection of the two ellipses given by their equations as follows:
\[ \dfrac{x^2}{16} + \dfrac{(y + 1)^2} { 4} = 1 \]
\[ \dfrac{x^2}{2} + \dfrac{(y + 2)^2}{12} = 1 \]
__Solution to Example 1:__
We first multiply all terms of the first equation by \( 16 \) and all the terms of the second equation by \( - 2 \) and simplify to obtain equivalent equations given by:

\( x^2 + 4 (y + 1)^2 = 16 \)

\( - x^2 - \dfrac{1}{6} (y + 2)^2 = - 2 \)

We now add side by side the two equations to obtain a quadratic equation

\( 4 (y + 1)^2 - \dfrac{1}{6} (y + 2)^2 = 14 \)

Multiply all terms by 6, group like terms and rewrite the equation as

\( 23 y^2 + 44y - 64 = 0 \)

Solve the quadratic equation for \( y \) to obtain two solutions

\( y \approx 0.97 \) and \( y \approx -2.88 \)

We now substitute the values of \( y \) already obtained into the equation \( x^2 + 4 (y + 1)^2 = 16 \)
and solve it for \( x \) to obtain the \( x \) values

for \( y \approx 0.97 \); \( x \) values are given by: \( x \approx 0.730365 \) and \( x \approx -0.730365 \)

for \( y \approx -2.88 \); \( x \) values are given by: \( x \approx 1.36788 \) and \( x \approx -1.36788 \)

The 4 points of intersection of the two ellipses are

\( ( 0.730365 , 0.97) \); \( ( -0.73 , 0.97) \); \( (1.37 , -2.88) \); \( (- 1.36788 , -2.88) \)

The graph of the two ellipses given above by their equations are shown below with their points of intersection.

## More Links and References on Ellipses

Points of Intersection of an Ellipse and a line

Find the Points of Intersection of a Circle and an Ellipse

Equation of Ellipse, Problems.

College Algebra Problems With Answers - sample 8: Equation of Ellipse

HTML5 Applet to Explore Equations of Ellipses

Ellipse Area and Perimeter Calculator