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 = 2x - 2

  • We now substitute y by 2x - 2 in the equation of the ellipse
    x2 / 9 + (2x - 2)2 / 4 = 1

  • Multiply all terms by 36, group like terms and rewrite the equation as
    40x2 - 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.

Points of intersection of an ellipse and a line



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