Deriving the General Equation of a Circle: A Step-by-Step Explanation

The general equation of a circle is derived from the Pythagorean theorem.

Consider a circle with center (h, k) and radius r. Let (x, y) be any point on the circle.

Using the distance formula, we know that the distance between (h, k) and (x, y) is:

sqrt((x-h)^2 + (y-k)^2)

Since (x, y) is on the circle, its distance from the center is the same as the radius r:

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

Squaring both sides and simplifying, we get:

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

This is the general equation of a circle, where (h, k) represents the center and r represents the radius.