# Points of Intersection of an Ellipse and a line

This is tutorial on finding the points of intersection of an ellipse and a line given by their equations.

## Example 1

Find the points of intersection of an ellipse and a line given by their equations as follows:
$$\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$$
$$y - 2x = -2$$
Solution to Example 1:
We first solve the equation of the line for $$y$$ to obtain:
$$y = 2x - 2$$
We now substitute $$y$$ by $$2x - 2$$ in the equation of the ellipse
$$\dfrac{x^2}{9} + \dfrac{(2x - 2 )^2}{4} = 1$$
Multiply all terms by 36, group like terms and rewrite the equation as
$$40 x^2 - 72 x = 0$$
Solve the quadratic equation for $$x$$ to obtain two solutions
$$x = 0$$ and $$x = \dfrac{9}{5}$$
We now substitute the values of $$x$$ already obtained into the equation $$y = 2x - 2$$ and find $$y$$
for $$x = 0$$, $$y = -2$$ and for $$x = \dfrac{9}{5}$$, $$y = \dfrac{8}{5}$$
There 2 points of intersection given by
$$(0, -2)$$ and $$\left(\dfrac{9}{5}, \dfrac{8}{5}\right)$$

The graphs of the ellipse and the line given by their equations above and their points of intersection are shown below.

### More Links and References on Ellipses

Find the Points of Intersection of two Ellipses
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