How do you graph #r=10costheta#?

1 Answer
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 = 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]}