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:
x^{2} / 9 + y^{2} / 4 = 1
y  2x = 2Solution 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
x^{2} / 9 + (2x  2)^{2} / 4 = 1

Multiply all terms by 36, group like terms and rewrite the equation as
40x^{2}  72x = 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)
Shown below is the graph of the ellipse, the line and their points of intersection.
More links and references related to the above topics.
