Question

The centre of the circle passing through points $(0,0)(1,0)$ and touching the circle $x^2+y^2=9$, is

• A. $\left(\frac{3}{2},\frac{1}{2}\right)$
• B. $\left(\frac{1}{2},\frac{3}{2}\right)$
• C. $\left(\frac{1}{2},\frac{1}{2}\right)$
• D. $\left(2^{1/2},-2^{1/2}\right)$

Answer 1

The correct option is D $\left(2^{1/2},-2^{1/2}\right)$

Equation of circle passes through the points $(0,0)$ and $(1,0)$ is $x^2+y^2-x+2fy=0$
Since, $C_1{C}_2=r_2-r_1$
$\therefore \sqrt{\frac{1}{4}+f^2}=3-\sqrt{\frac{1}{4}+f^2}$
$\Rightarrow f^2+\frac{1}{4}=\frac{9}{4}$
$\Rightarrow f^2=2$
$\Rightarrow f=\pm \sqrt{2}$
$\therefore$ Centre is $(\sqrt{2},-\sqrt{2})$.