# 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