Proof of $r=a \cos(\theta)$

Question

What is the proof that the polar equation $r=a \cos(\theta)$ is a circle of diameter a? An intuitive explanation would also work. What are the axis labeled, $\theta$ and x or y and x? I cannot grasp (intuitively understand) how the points are determined to form the curve.

Answer 1

Every angle on a point on a circle subtended by the diameter is $90^\circ$. Therefore the diameter, of length $a$ is the hypotenuse, and the distance $r$ is adjacent to the angle $\theta$. Using the adjecent over hypotenuse rule, $\frac{r}{a}=\cos\theta$ and so $r=a\cos\theta$. Therefore a circle of diameter a is a solution of the polar equation.

Answer 2

For $r>0:$

$$r=a\cos(\theta) \iff r^2=ar\cos(\theta) \iff x^2+y^2= ax \iff \Big(x-\frac{a}{2}\Big)^2+y^2=\frac{a^2}{4}$$