Question

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

Answer: (B)

$(\frac{1}{2},-\sqrt{2})$

Let the equation of circle be -

$\Rightarrow x^2+y^2+2gx+2fy+c=0$

Passes through $(0,0)\to c=0$

Passes through $(1,0)\to g=\frac{-1}{2}$

Now, $(0,0)$ lies inside $x^2+y^2=9$

$\therefore$ Circles touch internally.

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

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

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

$\Rightarrow g^2+f^2=\frac{9}{4}$

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

$\Rightarrow f=-\sqrt{2}$

$\therefore$ Centre $(\frac{1}{2},-\sqrt{2})$

Hence, solved.