Question on polar coordinates and cartesian coordinates I know the conversion between polar coordinates and cartesian coordinates. Nevertheless, I cannot understand why $r=2a\cos\theta$ represents a circle of radius $a$ and center $(a,0)$.
Can anyone explain it?
The graph hopefully makes clear angle subtended at origin is half that at center inside semi-circle and placed that way.

$ r = 2 a \cos \theta $

where the radius vector $r$ is a projection of diameter $ 2 a. $

The polar equation of all circles through the origin, rotated by $\alpha $ is:

$ r = 2 a \cos (\theta + \alpha) $
I suggest you draw out the graph with the circle because that's how I understand it. You could try and put it a few values for $\theta$ to see how the shape of the circle comes out.