# How do you graph r=10costheta?

Sep 24, 2016

The graph is the circle of radius 5 with center at (5, 0) on the initial line $\theta = 0$. This passes through the pole r = 0..

#### Explanation:

The polar equation of the family of circles through the pole r = 0)

and center at (a. 0) is

$r = 2 a \cos \theta$. The radius a is the parameter for the family.

So, here, $r = 10 \cos \theta$ represents a member of this family,

with parameter a = 5.

The general polar equation of the grand family of all circles, with

center at Cartesian $\left(\alpha , \beta\right)$ and radius 'a' is

( from ${\left(x - \alpha\right)}^{2} + {\left(y - \beta\right)}^{2} = {a}^{2}$)

r = alpha cos theta + beta sin theta +-sqrt(((alpha cos theta +beta sin theta)^2- (alpha^2+beta^2-a^2))

As $r \ge 0$, negative sign is for ${\alpha}^{2} + {\beta}^{2} > {a}^{2}$, when

the pole r = 0 is outside the circle.

Easy-to-remember direct polar form is

${a}^{2} = {r}^{2} - 2 r b \cos \left(\theta - \gamma\right) + {b}^{2}$,

with the center at polar $\left(b , \gamma\right)$ and radius 'a'.

In this example $r = 10 \cos \theta$,

Cartesian $\alpha = 5 , \beta = 0$.

Radius a = 5, and in the polar coordinates ,

the center is $\left(b , \gamma\right)$ = (5, 0) .

Graph of $r = 10 \cos \theta$:
graph{x^2+y^2 -10x = 0[-11 11 -5.5 5.5]}