Question

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

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

Answer 1

Answer: (D)

Let the equation of the circle be
$x^2+y^2+2gx+2fy+c=0$
which passes through $(0,0)$ and $(1,0)$.
$\therefore c=0$ and $1+2g+c=0$
$\Rightarrow c=0,g=-\frac{1}{2}$
The circle $x^2+y^2+2gx+2fy+c=0$
touches the circle $x^2+y^2=9$.
$\therefore \sqrt{g^2+f^2}=3\pm \sqrt{g^2+f^2-c}$
$\Rightarrow \sqrt{\frac{1}{4}+f^2}=3\pm \sqrt{\frac{1}{4}+f^2}$
$\Rightarrow 3=2\sqrt{\frac{1}{4}+f^2}$
$\Rightarrow f^2=\frac{9}{4}-\frac{1}{4}=2$
$\Rightarrow f=\pm \sqrt{2}$
Hence, the coordinates of the centre are
$\left(\frac{1}{2},\pm \sqrt{2}\right).$