Question

Let $S_n$ be the director circle of the circle $S_{n-1}$, where $n\ge 2$. $S_1:x^2+y^2=1$. The equation of the circle orthogonal to the circle $S_{10}$ is given by $x^2+y^2+2gx+2fy+c=0$. Then

• A. the value of $c$ cannot be determined
• B. no such circle is possible
• C. there is a unique value of $c$
• D. the value of $c$ depends upon $g$ and $f$

Answer 1

The correct option is C there is a unique value of $c$

$S_1:x^2+y^2=1S_2:x^2+y^2=2S_3:x^2+y^2=2^2\therefore S_n:x^2+y^2=2^{n-1}\Rightarrow S_{10}:x^2+y^2=2^9$

$x^2+y^2+2g_{1}x+2f_{1}y+c_1=0$
$x^2+y^2+2g_{2}x+2f_{2}y+c_2=0$
Condition of orthogonal is,
$2g_1{g}_2+2f_1{f}_2=c_1+c_2$

$\therefore 2(0+0)=c-2^9$
$\Rightarrow c=512$