As an application of completing the square, I'll look at equations of circles. The standard form for the equation of a circle with radius r and center is

To see this, look at the right triangle below.

The horizontal side is the difference between the x-coordinates of and , which is . I need to take the absolute value in case is to the left of .

Likewise, the vertical side is the difference between the y-coordinates of and , which is . I need to take the absolute value in case is below .

By Pythagoras' Theorem,

Since squares are nonnegative, I can drop the absolute values and get

* Example.* (a) Find the center and radius of the
circle whose equation is

The center is and the radius is 6.

(b) What is the equation of the circle whose center is and whose radius is 1.1?

(Don't multiply out the left side --- leave it as is.)

You can find the center and radius of a circle whose equation is not in standard form by completing the square.

* Example.* Find the center and radius of the
circle

The center is and the radius is 8.

* Example.* Find the center and radius of the
circle

To complete the square in x, I have , and . To complete the square in y, I have , and . So I have

The center is and the radius is .

Copyright 2016 by Bruce Ikenaga