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
( 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.
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