What is the equation of a circle with center at (h, k) and radius r?

The equation of a circle is derived from the definition of a circle in a Cartesian coordinate system, which is the set of all points that are equidistant from a fixed point known as the center. The center of the circle is represented by the coordinates (h, k), and the distance from the center to any point on the circle is the radius, denoted as r.

To formulate this mathematically, consider any point (x, y) on the circle. The distance from the point (x, y) to the center (h, k) can be expressed using the distance formula, which states:

[

\text{Distance} = \sqrt{(x - h)^2 + (y - k)^2}

]

For a point to be on the circle, this distance must equal the radius r. Therefore, we set up the equation:

[

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

]

Squaring both sides eliminates the square root, resulting in:

[

(x - h)^2 + (y - k)^2 = r^2

]

This equation accurately represents all points (x, y) that are at a distance r from