If $S : x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ intersects both the lines $x y - 3 x = 0$ orthogonally and touches
The circle $x^{2} + y^{2} - 6 x - 12 y + 36 = 0$ externally, then

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$S$ neither intersects nor touches the x-axis

radius of the circle $S$ lies in (1,2)

radius of the circle $S$ lies in (2,3)

equation of the transverse common tangent to both the circle is $x + y - 3 \sqrt{2} - 9 = 0$

answer is A, B.

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Detailed Solution

$x (y - 3) = 0$
$\Rightarrow (0, 3)$ is the centre
$C_{1} C_{2} = r_{1} + r_{2}$

$\sqrt{9 + 9} = r + 3$
$\Rightarrow r = 3 (\sqrt{2} - 1) ≃ 1.2$
Transverse common tangent to both the circle
   $x + y + k = 0$
Applying    $p = r \Rightarrow k = \pm 3 \sqrt{2} - 9$
$x + y + 3 \sqrt{2} - 9 = 0$ or    $x + y - 3 \sqrt{2} - 9 = 0$ (rejected)