# 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, is presented.

### Example 1

Find the points of intersection of the circles given by their equations as follows \[ (x - 2)^2 + (y - 3)^2 = 9 \] \[ (x - 1)^2 + (y + 1)^2 = 16 \]### Solution to Example 1

The points of intersections are found by solving the above system of equations.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 above quadratic equation for \( y \) to obtain two solutions

\( y = \frac{31 + 8\sqrt{2}}{17} \approx 2.49 \)

and \( y = \frac{31 - 8\sqrt{2}}{17} \approx 1.16 \)

We now substitute the values of \( y \) already obtained into the equation \( x = 9 - 4y \) to obtain the values for \( x \) as follows

\( x = \frac{29 + 32\sqrt{2}}{17} \approx - 0.96 \)

and \( x = \frac{29 - 32\sqrt{2}}{17} \approx 4.37 \)

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 Solvers Points 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