Question

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

• A. $(3/2,1/2)$
• B. $(1/2,3/2)$
• C. $(1/2,1/2)$
• D. $(1/2,-2)$

Answer 1

The correct option is D $(1/2,-2)$

The circle must be touching $x^2+y^2=g$ internally.

Let $S:x^2+y^2+2gx+2fy+c=0$ be the equation of the circle.

$S(0,0)=0$

$⟹C=0$

$S(1,0)=0$

$⟹1+2g+c=0⟹g=\frac{-1}{2}$

Since the circles touch internally.

$|r_1–r_2|=C_1{C}_2$

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

$⟹\sqrt{f^2+\frac{1}{4}}=3-\sqrt{f^2+\frac{1}{4}}$

$⟹f^2+\frac{1}{4}=\frac{g}{4}⟹f=\pm 2$

Center $=(-g,-f)=(\frac{1}{2},\pm 2)$