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.