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Equation of a Circle

A circle is the set of all points in a plane at a given distance (called the radius ) from a given point (called the center.)

A line segment connecting two points on the circle and going through the center is called a diameter of the circle.

Assume that $\left(x,y\right)$ are the coordinates of a point on the circle shown. The center is at $\left(h,k\right)$ , and the radius is $r$ .

Use the Distance Formula to find the equation of the circle.

$\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}=d$

Substitute $\left(x_1,y_1\right)=\left(h,k\right),\left(x_2,y_2\right)=\left(x,y\right)$ and $d=r$ .

$\sqrt{{\left(x-h\right)}^2+{\left(y-k\right)}^2}=r$

Square each side.

${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$

The equation of a circle with center $\left(h,k\right)$ and radius $r$ units is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ .