# 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 1Find the points of intersection of an ellipse and a line given by their equations as follows:^{2} / 9 + y^{2} / 4 = 1
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 EllipsesFind the Points of Intersection of two EllipsesFind 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 |