Question

If two circles $x^2+y^2-6x-12y+1=0and{x}^2+y^2-4x-2y-11=0$ cut a third circle orthogonally then the radical axis of the two circles passes through

• A. $(1,1)$
• B. $(0,6)$
• C. centre of the third circle
• D. mid-point of the line joining the centres of the given circles.

Answer 1

Answer: (A), (C)

The correct options are
A $(1,1)$
C centre of the third circle

Radical axis of the given circle is $x+5y-6=0$ which passes through $(1,1)$
Let the given circles intersect the circle
$x^2+y^2+2gx+2fy+c=0$ orthogonality then
$2g(-3)+2f(-6)=c+1$
$2g(-2)+2f(-1)=c-11\Rightarrow 2g(-1)+2f(-1)=c-12\Rightarrow -g-5f-6=0$
$\Rightarrow$ the radical axis passes through the center $(-g,-f)$ of the third circle.