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) \[x=\cos \theta -1,y=\frac{\sqrt{3}}{2}+\sin \theta \]
Correct Answer: B
Solution :
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.
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