The parametric equations of the circle $x^2+y^2-a x-b y=0$ are

The parametric equations of the circle $x^2+y^2-a x-b y=0$ are
  1. $x=\frac{\mathrm{a}}{2}+\frac{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}{2} \cos \theta, y=\frac{\mathrm{b}}{2}+\frac{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}{2} \sin \theta$
  2. $x=\frac{-\mathrm{a}}{2}+\frac{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}{4} \sin \theta, y=\frac{-\mathrm{b}}{2}+\frac{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}{4} \cos \theta$
  3. $x=\frac{\mathrm{a}}{2}+\sqrt{\frac{\mathrm{a}^2+\mathrm{b}^2}{2}} \sin \theta, y=\frac{\mathrm{b}}{2}+\sqrt{\frac{\mathrm{a}^2+\mathrm{b}^2}{2}} \cos \theta$
  4. $x=\frac{\mathrm{a}}{2}+\frac{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}{4} \cos \theta, y=\frac{\mathrm{b}}{2}+\frac{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}{4} \sin \theta$

Solution

Given equation can be written as $\begin{aligned} & \quad\left(x^2-\mathrm{a} x+\frac{\mathrm{a}^2}{4}\right)+\left(y^2-\mathrm{b} y+\frac{\mathrm{b}^2}{4}\right)=\frac{\mathrm{a}^2}{4}+\frac{\mathrm{b}^2}{4} \\ & \quad \Rightarrow\left(x-\frac{\mathrm{a}}{2}\right)^2+\left(y-\frac{\mathrm{b}}{2}\right)^2=\left(\sqrt{\frac{\mathrm{a}^2+\mathrm{b}^2}{4}}\right)^2 \\ & \therefore \quad \mathrm{~h}=\frac{\mathrm{a}}{2}, \mathrm{k}=\frac{\mathrm{b}}{2} \text { and } \mathrm{r}=\sqrt{\frac{\mathrm{a}^2+\mathrm{b}^2}{4}} \end{aligned}$ $\therefore \quad$ Parametric equations of circle are $x=\frac{a}{2}+\frac{\sqrt{a^2+b^2}}{2} \cos \theta$ and $y=\frac{\mathrm{b}}{2}+\frac{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}{2} \sin \theta$

Asked in: MHT CET 2024 (11 May Shift 2)

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