Question

The centerof the circle passing through $(0,0)$ and $(1,0)$ and touching the circle $y^2=9$ is

• A. $(\frac{1}{2},\frac{1}{2})$
• B. $(\frac{1}{2},-\sqrt{2})$
• C. $(\frac{3}{2},\frac{1}{2})$
• D. $(\frac{1}{2},\frac{3}{2})$

Answer 1

The correct option is B $(\frac{1}{2},-\sqrt{2})$

Let the required circle be $x^2+y^2+2gx+2fy+c=0$

Since, it passes through points $(0,0)$ & $(1,0)$

$\Rightarrow c=0$ and $g=\frac{-1}{2}$

Points(0,0) and (1,0) lie inside the circle $x^2+y^2=9$. So, two circles touch internally.

$\Rightarrow c_1{c}_2=r_1-r_2$

$\sqrt{g^2+y^2}=3-\sqrt{g^2+y^2}$

$\Rightarrow \sqrt{g^2+y^2}=\frac{3}{2}$

$\Rightarrow y^2=\frac{9}{4}-\frac{1}{4}=2$

Hence, the center of circle $=(-g,-f)=(\frac{1}{2},-\sqrt{2})$