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: x² / 9 + y² / 4 = 1 y - 2x = -2 Solution to Example 1:
We first solve the equation of the line for y to obtain:
y = 2 x - 2
We now substitute y by 2x - 2 in the equation of the ellipse
x² / 9 + (2 x - 2)² / 4 = 1
Multiply all terms by 36, group like terms and rewrite the equation as
40 x² - 72 x = 0
Solve the quadratic equation for x to obtain two solutions
x = 0 and x = 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 = 9/5, y = 8/5
There 2 points of intersection given by
( 0 , -2) and (9/5 , 8/5)

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