What is the Equation of a Circle, Definition and Formulas:

In this article, we will discuss the equation of the circle also the standard equation of a circle in an easy way. The equation of a circle comes in two forms. If the equation of a circle is in standard form, we can easily identify the center of the circle $(h,k)$ and radius $r$

The set of all points in the plane that are equally distance from a fixed point is called a circle. The fixed point is called the center of the circle and the distance from the center of the circle to any point on the circle is called the radius of the circle.
If $C(h,k)$ is centre of a circle, $r$ its radius and $P(x, y)$ any point on the circle, then the circle,
denoted $S(C ; r)$ in set notation is

$S(c;r)={P(x,y)\;\;\left|\overline{CP}\right|=r}$

By the distance formula, we get

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

or $\;\;(x-h)^2+(y-k)^2=r^2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$

is an equation of circle in standard form.

If the center of the circle is the origin, then $(1)$ reduces to

$\left|\overline{CP}\right|=\sqrt{x^2+y^2}=r^2$

If $r = 0$ , the circle is called a point circle which consists of the Centre only.
Let $P(x, y)$ be any point on the circle $(2)$ and let the inclination of $OP$ be q as shown in the figure. It is clear that

$\begin{array}{r}x=r\;\cos\theta\;\;,\;\;\;y=r\;\sin\theta\end{array}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3)$

The point $P(r\cos\theta,r\sin\theta)$ lies on $(2)$ for all values of $\theta$ . Equations $(3)$ are called parametric equations of the
circle $(2)$ .