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
x
2 / 9 + (2 x - 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 = 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. 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