Find the equation of the circle whose center is $\left(5, \dfrac{\pi}{3}\right)$ and radius $7$.
Given: $\;\;$ Center of the circle $= \left(R, \phi\right) = \left(5, \dfrac{\pi}{3}\right)$
i.e. $\;$ $R = 5, \;\; \phi = \dfrac{\pi}{3}$
Radius of the circle $= a = 7$
Let the equation of the required circle be
$R^2 + r^2 - 2 R r \cos \left(\theta - \phi\right) = a^2$
Substituting the values of $R$, $\phi$ and $a$, we get
$25 + r^2 - 2 \times 5 \times r \cos \left(\theta - \dfrac{\pi}{3}\right) = 49$
i.e. $\;$ $r^2 - 10r \left(\cos \theta \cos \dfrac{\pi}{3} + \sin \theta \sin \dfrac{\pi}{3}\right) - 24 = 0$
i.e. $\;$ $r^2 - 10r \left(\dfrac{1}{2} \cos \theta + \dfrac{\sqrt{3}}{2} \sin \theta\right) - 24 = 0$
i.e. $\;$ $r^2 - 5r \left(\cos \theta + \sqrt{3} \sin \theta\right) - 24 = 0$