If a circle $S$ passing through the point $(3,4)$ cuts the circle $x^2+y^2=36$ orthogonally, then the locus of the centre of $S$ is

• A

  $x^2+y^2-6 x-8 y+11=0$
• Correct answer
  B

  $6 x+8 y-61=0$
• C

  $x^2+y^2-8 x-6 y+11=0$
• D

  $6 x+8 y+11=0$

Solution

Let the circle is $x^2+y^2+2 g x+2 f y+c=0$, having centre $(-g,-f)$, since it passes through the point $(3,4)$
And circle is intersecting the other circle $$ \begin{aligned} & x^2+y^2=36 \text { orthogonally, so } \\ & \qquad 2 g(0)+2 f(0)=c-36 \end{aligned} $$
From Eqs. (i) and (ii) $$ -6 g-8 f=61, $$ Now, on taking locus of point $(-g,-f)$, we are getting $6 x+8 y-61=0$.

Hence, the correct answer is option: 'B'.