Circle 3 Question 15

15. The equation of circle $C$ is

(a) $(x-2 \sqrt{3})^{2}+(y-1)^{2}=1$

(b) $(x-2 \sqrt{3})^{2}+y+\frac{1}{2}^{2}=1$

(c) $(x-\sqrt{3})^{2}+(y+1)^{2}=1$

(d) $(x-\sqrt{3})^{2}+(y-1)^{2}=1$

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Answer:

Correct Answer: 15. (d)

Solution:

  1. Let centre of circle $C$ be $(h, k)$.

$$ \begin{aligned} & \text { Then, } \\ & \left|\frac{\sqrt{3} h+k-6}{\sqrt{3+1}}\right|=1 \\ & \begin{array}{lc} \Rightarrow & \sqrt{3} h+k-6=2,-2 \\ \Rightarrow & \sqrt{3} h+k=4 \end{array} \end{aligned} $$

[rejecting 2 because origin and centre of $C$ are on the same side of $PQ$ ] The point $(\sqrt{3}, 1)$ satisfies Eq. (i).

$\therefore \quad$ Equation of circle $C$ is $(x-\sqrt{3})^{2}+(y-1)^{2}=1$.

Clearly, points $E$ and $F$ satisfy the equations given in option (d).



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