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}