# Find the Points of Intersection of Two Ellipses

 This is tutorial on finding the points of intersection of two ellipses given by their equations. Example 1: Find the points of intersection of the two ellipses given by their equations as follows: x2 / 16 + (y + 1)2 / 4 = 1 x2 / 2 + (y + 2)2 / 12 = 1 Solution to Example 1: We first multiply all terms of the first equation by 16 and all the terms of the second equation by - 2 to obtain equivalent equations: x2 + 4 (y + 1)2 = 16 - x2 - (1/6) (y + 2)2 = - 2 We now add the same sides of the two equations to obtain a quadratic equation 4 (y + 1)2 - (1 / 6) (y + 2)2 = 14 Multiply all terms by 6, group like terms and rewrite the equation as 23 y2 + 44y - 64 = 0 Solve the quadratic equation for y to obtain two solutions y ≈ 0.97 and y ≈ -2.88 We now substitute the values of y already obtained into the equation x2 + 4 (y + 1)2 = 16 and solve it for x to obatain the x values for y ≈ 0.97 ; x values are given by: x ≈ 0.730365 and x ≈ -0.730365 for y ≈ -2.88 ; x values are given by: x ≈ 1.36788 and x ≈ -1.36788 The 4 points of intersection of the two ellipses are ( 0.730365 , 0.97) ; ( -0.73 , 0.97) ; (1.37 , -2.88) ; (- 1.36788 , -2.88) Shown below is the graph of two ellipses and their points of intersection. More links and references related to the above topics. Equation of Ellipse, Problems.