Find the Points of Intersection of a circle and an ellipse
A tutorial on finding the points of intersection of a circle and an ellipse given by their equations.
Example 1
Find the points of intersection of the circle and the ellipse given by their equations as follows: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 links
Tutorials on equation of circle.Tutorials on equation of circle (2).
Interactive tutorial on equation of circle. Computer Technology Simply Explained