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

Feb 4, 2015

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

$6 \cos \left(\theta\right) = 2 + 2 \cos \left(\theta\right)$

And solving for $\theta$ you get:

$6 \cos \left(\theta\right) - 2 \cos \left(\theta\right) = 2$

$4 \cos \left(\theta\right) = 2$

$\cos \left(\theta\right) = \frac{1}{2}$ which is valid:

for $\theta = \frac{\pi}{3}$ and $\theta = 5 \frac{\pi}{3}$

Substituting back you get:
$r = 6 \cos \left(\theta\right) = 6 \cdot \frac{1}{2} = 3$
or
$r = 2 + 2 \cos \left(\theta\right) = 2 + 2 \cdot \frac{1}{2} = 3$ again.
Giving 2 points of intersection:
$\left(3 , \frac{\pi}{3}\right)$
$\left(3 , \frac{5}{3} \pi\right)$
Graphically:

hope it helps