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.

Points of intersection of an ellipse and a line

Points of Intersection of an Ellipse and a line

Example 1

More Links and References on Ellipses