Question

The parametric equations of the circle $x^2+y^2+x+\sqrt{3}y=0$ are

• A. $x=1+\cos \theta ,y=\frac{\sqrt{3}}{2}+\sin \theta$
• B. $x=-\frac{1}{2}+\cos \theta ,y=-\frac{\sqrt{3}}{2}+\sin \theta$
• C. $x=\frac{1}{2}+\cos \theta ,y=-\frac{\sqrt{3}}{2}+\sin \theta$
• D. $x=\frac{1}{2}+\frac{1}{2}+\cos \theta ,y=\frac{\sqrt{3}}{2}+\frac{1}{2}+\sin \theta$
• E. $x=\cos \theta -1,y=\frac{\sqrt{3}}{2}+\sin \theta$

Answer 1

Answer: (B)

Equation of circle $x^2+y^2+x+\sqrt{3}y=0$
$\Rightarrow$ $(x^2+x)+(y^2+\sqrt{3}y)=0$
$\Rightarrow$ $\left(x^2+x+\frac{1}{4}\right)+\left(y^2+\sqrt{3}y+\frac{3}{4}\right)=\frac{1}{4}+\frac{3}{4}$
$\Rightarrow$ ${\left(x+\frac{1}{2}\right)}^2+{\left(y+\frac{\sqrt{3}}{2}\right)}^2=1$
$\Rightarrow$ ${\left(x-\left(\frac{-1}{2}\right)\right)}^2+{\left(y-\left(\frac{-\sqrt{3}}{2}\right)\right)}^2=1$ Let $x+\frac{1}{2}\cos \theta \Rightarrow x=-\frac{1}{2}+\cos \theta$ and $y+\frac{\sqrt{3}}{2}=\sin \theta \Rightarrow y=\frac{-\sqrt{3}}{2}+\sin \theta$
Which are the required parametric coordinates of the given circle.