Question

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

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

Answer 1

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

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 $x^2+y^2=9$
$\therefore \sqrt{g^2+f^2}=3\pm \sqrt{g^2+f^2-c}$
$[\because C_1{C}_2=I_1\pm I_2]$
$\Rightarrow \sqrt{\frac{1}{4}+f^2}=3\pm \sqrt{\frac{1}{4}+f^2}$
$\Rightarrow 3=2\sqrt{\frac{1}{4}+f^2}$
$\Rightarrow f^2=\frac{9}{4}-\frac{1}{4}=2$
$\Rightarrow f=\pm \sqrt{2}$
Hence, the coordinates of the centre are $(\frac{1}{2},\pm \sqrt{2})$.