# 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.

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