 # 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:

x2 + y2 = 4
x2 / 4 + (y - 1)2 / 9 = 1

### Solution to Example 1

• We first multiply all terms of the second equation by -4 to obtain:
x2 + y2 = 4
-x2 - (4 / 9) (y - 1)2 = - 4

• We now add the same sides of the two equations to obtain a linear equation
y2 - (4 / 9) (y - 1)2 = 0
• Which may be written as
5y2 + 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 x2 + y2 = 4 and solve it for x as follows
x2 + (-2)2 = 4
x = 0
• We now substitute the values of y = 2/5 already obtained into the equation x2 + y2 = 4 and solve it for x as follows
x2 + (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. 