Question

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

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

Answer 1

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

Let the required circle be

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

Since it passes through $(0,0)$ and $(1,0)$

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

Point $(0,0)$ and $(1,0)$ lie inside the circle $x^2+y^2=9$

So two circle touch internally

$C_1{C}_2=r_1-r_2$

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

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

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

Hence the centers of required circle are $(\frac{1}{2},-\sqrt{2})$