Find the Points of Intersection of two Circles
A tutorial on how to find the points of intersection of two circles given by their equations; general solution.
Example 1
Find the points of intersection of the circles given by their equations as follows(x  1)^{2} + (y + 1)^{2} = 16
Solution to Example 1

We first expand the two equations as follows:
x^{2}  4x + 4 + y^{2}  6y + 9 = 9
x^{2}  2x + 1 + y^{2} + 2y + 1 = 16

Multiply all terms in the first equation by 1 to obtain an equivalent equation and keep the second equation unchanged
x^{2} + 4x  4  y^{2} + 6y  9 = 9
x^{2}  2x + 1 + y^{2} + 2y + 1 = 16

We now add the same sides of the two equations to obtain a linear equation
2x  3 + 8y  8 = 7

Which may written as
x + 4y = 9 or x = 9  4y

We now substitute x by 9  4y
in the first equation to obtain
(9  4y)^{2}  4(9  4y) + 4 + y^{2}  6y + 9 = 9

Which may be written as
17y^{2} 62y + 49 = 0

Solve the quadratic equation for y to obtain two solutions
y = (31 + 8√2) / 17 ≈ 2.49
and y = (31  8√2) / 17 ≈ 1.16

We now substitute the values of y already obtained into the equation x = 9  4y to obtain the values for x as follows
x = (29 + 32√2) / 17 ≈  0.96
and x = (29  32√2) / 17 ≈ 4.37

The two points of intersection of the two circles are given by
( 0.96 , 2.49) and (4.37 , 1.16)
Shown below is the graph of the two circles and the linear equation x + 4y = 9 obtained above.
More References and links
Step by Step Maths Worksheets SolversPoints of Intersection of Two Circles  Calculator.
Tutorials on equation of circle.
Tutorials on equation of circle (2).
Interactive tutorial on equation of circle. Computer Technology Simply Explained