Question

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

Answer 1

(2, 0) and r = 0.

Explanation:

The two meet when `r=1+cos theta=2cos 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 [-pi, pi]`.

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

at (1,0) and the whole circle is drawn for `theta in [-pi/2, pi/2]`

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

The reason is that,,

for `r = 2 cos theta`, r = 0, when `theta = +-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.