Find the intersecting points of two ellipses?

Question

4x^2+y^2=4
4x^2+9y^2=36

Answer 1

subtract 1st equation from the second:
8y^2=32
so y^2=4, and y = +/- 2

plug back into 1st equation:
4x^2+4=4

4x^2=0
x=0

Answer 2

Two ellipses can have anywhere from 0 to 4 points of intersection. Because of the nature of these two ellipses, they intersect in exactly two points, (0,2) and (0,-2). The algebra to solve the equations and find these points was given by the first responder.