The equation of a circle passing through ( 3, - 6) and touching both the axes, is

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$x^{2} + y^{2} + 30 x + 30 y + 225 = 0$

$x^{2} + y^{2} + 30 x - 30 y + 225 = 0$

$x^{2} + y^{2} + 6 x - 6 y + 9 = 0$

$x^{2} + y^{2} - 6 x + 6 y + 9 = 0$

answer is A.

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Detailed Solution

Circles given in option $(a)$ passes through $(3, - 6)$ and touches both the axes.

ALITER The circle touching both the axes in the fourth quadrant is $x^{2} + y^{2} - 2 a x + 2 a y + a^{2} = 0, a > 0 .$

It passes through (3,-6).

$∴ 45 - 6 a - 12 a + a^{2} = 0 \Rightarrow a^{2} - 18 a + 45 = 0 \Rightarrow a = 3, 5$

So, the equations are

$x^{2} + y^{2} - 6 x + 6 y + 9 = 0$ and $x^{2} + y^{2} - 10 x + 10 y + 25 = 0$

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