Equation of Circle
Equation of circle of radius $R$, center at $(x_0,y_0)$

Equation of Circle of Radius $R$ Passing through Origin
where $(\theta,\alpha)$ are polar coordinates of any point on the circle and $(R,\alpha)$ are polar coordinates of the center of the circle.

Conics [Ellipse, Parabola or Hyperbola]

If a point $P$ moves so that its distance from a fixed point [called the focus] divided by its distance from a fixed line [called the directrix] is a constant $e$ [called the eccentricity], then the curve described by $P$ is called a conic [so-called because such curves can be obtained by intersecting a plane and a cone at different angles].
If the focus is chosen at origin $O$the equation of a conic in polar coordinates $(r, \theta)$ is, if $OQ=p$and $LM=D$,
$r=\frac{p}{1-\epsilon\cos\theta}=\frac{\epsilon D}{1-\epsilon\cos\theta}$
The conic is
(i) an ellipse if $\epsilon< 1$
(ii) a parabola if $\epsilon=1$
(iii) a hyperbola if $\epsilon> 1$.