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What Is the Equation of a Circle?

Circle with radius r

Rule

The Circle

The equation of a circle with its center at (x0,y0) and with radius r is given by

(x x0)2 + (y y 0)2 = r2

This means that all values for (x,y) that fulfill the equation are on the circle boundary.

To find the radius of the circle you can use this formula:

r = (x x0 )2 + (y y0 )2

Example 1

You are working with a circle with its center at (1, 2). The point (3, 5) is on the circle boundary. What is the radius of this circle?

Here you have (x0,y0) = (1, 2) and the point on the circle is (x,y) = (3, 5). The radius of the circle is therefore

r = (3 + 1)2 + (5 2)2 = 5

Example 2

Show that x2 + 6x + y2 2y 6 = 0 is a circle

To show this, you have to use the method to complete the square. You add and subtract (b 2 ) 2 for both the x-terms and the y-terms. You then get the following:

0 = x2 + 6x + y2 2y 6, = x2 + 6x + (6 2) 2 (6 2) 2 = + y2 2y + ( 2 2 ) 2 ( 2 2 ) 2 = 6 = x2 + 6x + 9 9 = + y2 2y + 1 1 6 = (x + 3)2 + (y 1)2 9 1 6 = (x + 3)2 + (y 1)2 16

0 = x2 + 6x + y2 2y 6 = x2 + 6x + (6 2) 2 (6 2) 2 + y2 2y + ( 2 2 ) 2 ( 2 2 ) 2 6 = x2 + 6x + 9 9 + y2 2y + 1 1 6 = (x + 3)2 + (y 1)2 9 1 6 = (x + 3)2 + (y 1)2 16

Thus,
(x + 3)2 + (y 1)2 = 42

You can see from this that x2 + 6x + y2 2y 6 = 0 is a circle with its center at (3, 1) and a radius r = 4.

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