# Circles

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 . Contact information

Copyright 2016 by Bruce Ikenaga