Question

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

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

Answer 1

The correct option is D $(\frac{1}{2},\pm \sqrt{2})$

Equation of circle passes through origin is given by,
$x^2+y^2+2gx+2fy=0$ (i)
Also it passes through $(1,0)\Rightarrow 1+2g=0\Rightarrow g=-\frac{1}{2}$
It also touches the given circle $\Rightarrow C_1{C}_2=r_2-r_1$
$\Rightarrow \sqrt{g^2+f^2}=3-\sqrt{g^2+f^2}\Rightarrow g^2=f^2=9/4$
$\Rightarrow f^2=\frac{9}{4}-\frac{1}{4}=2\Rightarrow g=\pm \sqrt{2}$
Hence centre of the required circle is $(-g,-f)=(\frac{1}{2},\pm \sqrt{2})$