Question

How do you find the points of intersection of the curves with polar equations `r=6costheta` and `r=2+2costheta`?

Answer 1

I would substitute the first equation into the second for `r`, giving:

`6cos(theta)=2+2cos(theta)`

And solving for `theta` you get:

`6cos(theta)-2cos(theta)=2`

`4cos(theta)=2`

`cos(theta)=1/2` which is valid:

for `theta=pi/3` and `theta=5pi/3`

Substituting back you get:
`r=6cos(theta)=6*1/2=3`
or
`r=2+2cos(theta)=2+2*1/2=3` again.
Giving 2 points of intersection:
`(3,pi/3)`
`(3,5/3pi)`
Graphically:

hope it helps