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# For what relation will the circle touch each other $S_1=x^2+y^2+2gx+2fy=0$ and $S_2=x^2+y^2+2g'x+2f'y=0$?

$\begin{array}{1 1}(a)\;gf'=g'f\\(b)\;gf'=-g'f\\(c)\;g'f'=g'f\\(d)\;gf'=g'ff'\end{array}$

If two circle touch each other then distance between their centres=Sum or difference of their radii
For $S_1$
$C_1(-g,-f),r_1=\sqrt{g^2+f^2}$
For $S_2$
$C_2(-g',-f'),r_1=\sqrt{g'^2+f'^2}$
$\sqrt{(g-g')^2+(f-f')^2}=\sqrt{g^2+f^2}\pm \sqrt{g'^2+f'^2}$
$\sqrt{(g^2+f^2+g'^2+f'^2-2gg'-2ff')}=\sqrt{g^2+f^2}\pm \sqrt{g'^2+f'^2}$
Squaring $(gg'+ff')=\pm \sqrt{(g^2+f^2)(g'^2+f'^2)}$
Squaring
$g^2f'^2+f^2g'^2-2gg'ff'=0$
$(gf'=g'f)$
Hence (a) is the correct answer.