# How do you graph polar curves to see the points of intersection of the curves?

May 18, 2016

If the polar equations are $r = f \left(\theta\right) \mathmr{and} r = g \left(\theta\right)$ or, inversely, $\theta = {f}^{- 1} \left(r\right) \mathmr{and} \theta = {g}^{- 1} \left(r\right)$. eliminate either r or $\theta$, solve and substitute in one of the equations..

#### Explanation:

Explication:

Find the points of intersection of the cardioid

$r = a \left(1 + \cos \theta\right)$ and the circle r = a.

Eliminate r.

The equation for $\theta$ at a point of intersection is

$a = a \left(1 + \cos \theta\right)$. this is cos theta = 0 rarr theta = pi/2 and

(3pi)/2.

The common points are $\left(a , \frac{\pi}{2}\right) \mathmr{and} \left(a , \frac{3 \pi}{2}\right)$

For the graph, a = 1. Use $\left(x , y\right) = r \left(\cos \theta , \sin \theta\right)$

graph{(x^2+y^2-(x^2+y^2)^0.5-x)(x^2+y^2-1)=0[-2 4 -1.5 1.5]}

The two parabolas $1 = r \left(1 + \cos \theta\right) \mathmr{and} 1 = r \left(1 - \cos \theta\right)$

intersect at $\left(1 , \frac{\pi}{2}\right)$ and $\left(1 , 3 \frac{\pi}{2}\right)$.
graph{(x+(x^2+y^2)^0.5-1)(-x+(x^2+y^2)^0.5-1)=0[-3 3 -1.5 1.5]}