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Q.
Equation of the circle cutting orthogonally the three circles $x^2+y^2-2 x+3 y-7=0$, $x^2+y^2+5 x-5 y+9=0$ and $x^2+y^2+7 x-9 y+29=0$ is
Conic Sections
Solution:
$S _1- S _2=0 \Rightarrow 7 x -8 y +16=0 $
$S _2- S _3=0 \Rightarrow 2 x -4 y +20=0 $
$S _3- S _1=0 \Rightarrow 9 x -12 y +36=0$
On solving centre $(8,9)$
Length of tangent
$ =\sqrt{S_1}=\sqrt{64+81-16+27-7}=\sqrt{149}$
$ =(x-8)^2+(y-9)^2=149$
$ =x^2+y^2-16 x-18 y-4=0$