Equation of Circle

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

$(x-x_0)^2+(y-y_0)^2=R^2$

Equation of Circle of Radius $R$ Passing through Origin

$r=2R\cos(\theta-\alpha)$

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 $\epsilon> 1$.


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