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Q.
Two circles $S_{1}=x^{2}+y^{2}+2 g_{1} x+2 f_{1} y+c_{1}=0$ and
$S _{2}= x ^{2}+ y ^{2}+2 g _{2} x +2 f _{2} y + c _{2}=0$ cut each other orthogonally, then:
Conic Sections
Solution:
If two circles intersect at right angle i.e. the tangent at their point of intersection are at right angles, then the circles are called orthogonal circles.
The circles
$x^{2}+y^{2}+2 g x+2 f y+c=0$ and
$x^{2}+y^{2}+2 g_{1} x+2 f_{1} y+c_{1}=0$
are orthogonal, if: $2 gg _{1}+2 ff _{1}= c + c _{1}$
Thus, in the given question, the condition will be
$2 g_{1} g_{2}+2 f_{1} f_{2}=c_{1}+c_{2}$