# 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