Example 1
Find the points of intersection of the circle and the ellipse given by their equations as follows:
x^{2} + y^{2} = 4
x^{2} / 4 + (y  1)^{2} / 9 = 1
Solution to Example 1

We first multiply all terms of the second equation by 4 to obtain:
x^{2} + y^{2} = 4
x^{2}  (4 / 9) (y  1)^{2} =  4

We now add the same sides of the two equations to obtain a linear equation
y^{2}  (4 / 9) (y  1)^{2} = 0

Which may be written as
5y^{2} + 8y  4 = 0

Solve the quadratic equation for y to obtain two solutions
y = 2 and 2/5

We now substitute the values of y =  2 already obtained into the equation x^{2} + y^{2} = 4 and solve it for x as follows
x^{2} + (2)^{2} = 4
x = 0

We now substitute the values of y = 2/5 already obtained into the equation x^{2} + y^{2} = 4 and solve it for x as follows
x^{2} + (2/5)^{2} = 4
x = 4 √6 / 5 ≈ 1.96 and x =  4 √6 / 5 ≈ 1.96

The points of intersection of the ellipse and the circle are
(2 , 0) ; (4 √6 / 5 , 2/5) ; (4 √6 / 5 , 2/5)
Shown below is the graph of a circle and an ellipse and their points of intersection.
More References and linksTutorials on equation of circle.
Tutorials on equation of circle (2).
Interactive tutorial on equation of circle. 