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  4. The equation of the common tangent of the two touching circles y2 + x2 - 6x - 12y + 37 = 0 and x2 +y2 - 6y + 7 = 0 is

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Q. The equation of the common tangent of the two touching circles $y^2 + x^2 - 6x - 12y + 37 = 0$ and $ x^2 +y^2 - 6y + 7 = 0 $ is

KCETKCET 2006Conic Sections

Solution:

Let $S_{1} \equiv x^{2} +y^{2} -6x-12y+37=0$
and $ S_{2} \equiv x^{2}+y^{2}- 6y +7 = 0$
The equation of common tangent of the two circles is $S_1 - S_2 = 0$
$\Rightarrow x^{2} +y^{2}-6x-12y +37 $
$-\left(x^{2} +y^{2} -6y +7\right) =0 $
$ \Rightarrow -6x+6y+30 =0 $
$ \Rightarrow x-y -5 = 0 $