Question

How do you graph `r=10costheta`?

Answer 1

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 = 2a 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 `(alpha, beta)` and radius 'a' is

( from `(x-alpha)^2+(y-beta)^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 >=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 (theta-gamma) +b^2`,

with the center at polar `( b, gamma)` 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 `(b, gamma)` = (5, 0) .

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