Equation of Circle/Polar/Corollary

Corollary to Equation of Circle

Let $C$ be a circle whose radius is $R$.

Let $C$ be aligned in a polar coordinate frame such that its center is at the origin.

Then the equation of a $C$ is given by:

$r = R$

Proof

From Equation of Circle: Polar Form, we have a circle whose center is at $\polar {r_0, \varphi}$ whose radius is $R$ is:

$r^2 - 2 r r_0 \, \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$

So, when $\polar {r_0, \varphi} = \polar {0, 0}$:

\(\ds \sqrt {r^2 + 0^2 - 2 r \cdot 0 \cdot \map \cos {\theta - 0} }\)

\(=\)

\(\ds R\)

\(\ds \leadsto \ \ \)

\(\ds \sqrt{r^2}\)

\(=\)

\(\ds R\)

\(\ds \leadsto \ \ \)

\(\ds r\)

\(=\)

\(\ds R\)

$\blacksquare$