Equation of the circumference I: reduced equation

We will call the circumference: the geometric locus of points on the plane that are equidistant to a fixed point called center. This distance is named radius.

This property is the key to finding the analytical expression of a circumference.

Let's see it:

A circumference of center $C = ( a, b)$ and radius $r$, is formed by all the points $P = (x, y)$ whose distance to the center is $ r$.

Expressing this in a mathematical equation we have: $$\displaystyle d(C,P)=d((a,b),(x,y))= \sqrt{(x-a)^2+(y-b)^2} =r$$ Squaring this equation we obtain the reduced equation of the circumference: $$\displaystyle d(C,P)^2=\Big( \sqrt{(x-a)^2+(y-b)^2} \Big)^2=(x-a)^2+(y-b)^2=r^2$$ For which reason, any expression of the type $$(x-a)^2+(y-b)^2=r^2$$ is a circumference of radius $r$ and center at the point $(a, b)$.

$ (x-1)^2+(y-2)^2=3^2$ is a circumference of radius $3$ and centred at the point $(1, 2)$.

When we consider a circumference centred on the origin, we are taking $C = (0, 0)$ and therefore the equation is $x^2+y^2=r^2$.

$x^2+y^2=4^2$ it is centred on the origin and has a radius of $4$.

The circumference with center in the origin and radius $1$ is called a unit circumference.

If, for example, we want to write the equation of a circumference centred at point $(-8, 0)$ and with diameter $36$, the procedure is:

We calculate the radius: $$\displaystyle r=\frac{\mbox{diameter}}{2}=\frac{36}{2}=18$$

We replace the parameters in the equation of the circumference, with $r=18$ and $C = (-8, 0)$: $$\displaystyle (x-(-8)^2)+(y-0)^2=18^2 \Rightarrow (x+8)^2+y^2=18^2$$ So we already have the equation.