Welcome to the first lesson the Circle series.

This lesson will cover two topics – definition of the circle and derivation of the standard equation of a circle in the Cartesian plane.

To begin, let’s define the circle – A circle is a set of all points in a plane, which are equidistant from a fixed point. The fixed point is known as the **center** of the circle, and the (equal) distance between the center and any point on the circle is known as the **radius **of the circle.

In slightly technical words, a circle is the locus of a point, moving in a plane such that its distance from a given point is always constant.

Let’s go back to the Cartesian plane.

We’ll derive the standard equation to a circle – which has the center as origin (0, 0) and radius equal to ‘r’.

Let P(x, y) be any point on the circle.

Then, according to the definition of the circle, the distance between P and O will always be equal to ‘r’. That is OP = r.

We can write this using the distance formula as \( \sqrt{(x-0)^2+(y-0)^2} = r \)

On squaring both sides, we get the equation of the circle as **x ^{2} + y^{2} = r^{2}**. And that’s it!

The equation of the circle, having center as origin and radius ‘r’, is given by **x ^{2} + y^{2} = r^{2}**

In the next lesson, I’ll cover a few more variations of the equations to a circle.