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:

x^{2} / 16 + (y + 1)^{2} / 4 = 1

x^{2} / 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:
x^{2} + 4 (y + 1)^{2} = 16
- x^{2} - (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 y^{2} + 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 x^{2} + 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