Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (B)

$2$

Let $x^2+y^2+2gx+2fy=0$ is a circle possing through $(0,0)$
and also (1,0) lies on circle
$1+2g=0$, $\Rightarrow g=-\frac{1}{2}$
and this circle touches $x^2+y^2=9$ internally
so, $|r_1-r_2|=c_1{c}_2$
$3-\sqrt{g^2+f^2}=\sqrt{g^2+f^2}$
$g^2+f^2=\frac{9}{4}$
$f^2=\frac{9}{4}-\frac{1}{4}=2$
$f=\pm \sqrt{2}$
so, there are two such circles possible.