<s>[INST] Circle equation - diametric form - polar coordinates. A line segment joining $(a,\alpha)$, $(b,\beta)$ in polar coordinates is the diameter of a circle. I want to find the equation of this circle. It can be done by converting into the Cartesian system but I want to find the equation without moving out of polar. 
 [/INST] General equation of circle:
$$r^2-2rr_0\cos (\theta-\phi)+r_0^2=R^2$$
Now
\begin{align}
  (r_0 \cos \phi,r_0 \sin \phi) &=
  \left(
    \frac{a\cos \alpha+b\cos \beta}{2},
    \frac{a\sin \alpha+b\sin \beta}{2}
  \right) \\
  r_0^2 &=
  \left( \frac{a\cos \alpha+b\cos \beta}{2} \right)^2+
  \left( \frac{a\sin \alpha+b\sin \beta}{2} \right)^2 \\
  R^2 &=
  \left( \frac{a\cos \alpha-b\cos \beta}{2} \right)^2+
  \left( \frac{a\sin \alpha-b\sin \beta}{2} \right)^2
\end{align}
On simplifying,
\begin{align}
  r^2-r[a\cos (\theta-\alpha)+b\cos (\theta-\beta)]+
  ab\cos (\alpha-\beta) &=0 \\[5pt]
  [r-a\cos (\theta-\alpha)][r-b\cos (\theta-\beta)]+
  ab\sin (\theta-\alpha) \sin (\theta-\beta) &= 0
\end{align}
</s>