# How do you determine a point in common of r = 1 + cos theta and r = 2 cos theta?

Sep 28, 2016

(2, 0) and r = 0.

#### Explanation:

The two meet when $r = 1 + \cos \theta = 2 \cos \theta$.

Therein, $\cos \theta = 1$, and so, theta = 0#. The common r = 2.

The first equation $r = 1 + \cos \theta$ gives a full cardioid,

for $\theta \in \left[- \pi , \pi\right]$.

The second $r = 2 \cos \theta$ represents the unit circle with center

at (1,0) and the whole circle is drawn for $\theta \in \left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$

Interestingly, seemingly common point r=0 is not revealed here.

The reason is that,,

for $r = 2 \cos \theta$, r = 0, when $\theta = \pm \frac{\pi}{2}$

and for $r = 1 + \cos \theta$, r - 0, when theta = +-pi.

Thus the polar coordinate $\theta$ is not the same for both at r = 0.

This disambiguation is important to include, as we see, r = 0 as a

common point that is reached in different directions.

This is a reason for my calling r = 0 a null vector that has contextual

direction.