A **circle** is the set of points in a plane that are equidistant from one point. That one point is called the **center** of the circle, and the distance from it to any point on the circle is called the **radius** of the circle. The standard form for the equation of a circle with its center at (0, 0) and with a radius of length *r* is represented by the equation

*x* ^{2} + *y* ^{2} = *r* ^{2}

Example 1

Graph *x* ^{2} + *y* ^{2} = 16.

Recognize that *x* ^{2} + *y* ^{2} = 16 is the equation of a circle centered at (0, 0) with *r* ^{2} = 16. So *r* = 4, as shown in Figure 1.

Figure 1. Circle in standard position.

Example 2

Find the standard form for the equation of the circle centered (0, 0) with a radius of

The standard form for a circle centered at (0, 0) with a radius of *r* is

*x* ^{2} + *y* ^{2} = *r* ^{2}

Replacing *r* with

Therefore, *x* ^{2} + *y* ^{2} = 5 is the standard form of the equation of a circle centered at (0, 0) with a radius of

The standard form for a circle centered at ( *h, k*) with a radius of *r* is

( *x* – *h*) ^{2} + ( *y* – *k*) ^{2} = *r* ^{2}

Note that if ( *h, k*) = (0, 0) that would lead to

the equation for a circle centered at the origin.

Example 3

Graph the equation ( *x* – 3) ^{2} + ( *y* + 2) ^{2} = 25.

This equation represents a circle centered at (3, –2) with a radius of

Figure 2. Circle offset down and right.

Example 4

Find the standard form for the equation of the circle centered at (–6, 2) with a radius of

The standard form for the equation of a circle centered at ( *h, k*) with radius *r* is

( *x* – *h*) ^{2} + ( *y* – *k*) ^{2} = *r* ^{2}

Replacing *h* with –6, *k* with 2, and *r* with

Therefore, ( *x* + 6) ^{2} + ( *y* – 2) ^{2} = 18, which is the standard form of the equation of the circle centered at (–6, 2) with radius of

Example 5

From the equation given, find the center and radius for the following circle. Then graph the circle.

*x* ^{2} + *y* ^{2} – 8 *x* + 12 *y* – 12 = 0

This equation can be rewritten as

*x* ^{2} – 8 *x* + *y* ^{2} + 12 *y* = 12

Now, complete the square for each variable and add that amount to each side of the equation.

This circle is centered at (4, –6) with a radius of 8, as shown in Figure 3.

Figure 3. The graph of Example.