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Q.
If two circles, $x^{2}+y^{2}+2 g_{1} x+2 f_{1} y=0$ and $x^{2}+y^{2} +2 g_{2} x+2 f_{2} y=0$ touch each other externally then
Conic Sections
Solution:
If two circles touch each other, then
$C_{1} C_{2}=r_{1}+r_{2}$
$\Rightarrow \sqrt{\left(-g_{1}+g_{2}\right)^{2}+\left(-f_{1}+f_{2}\right)^{2}}=\sqrt{g_{1}^{2}+f_{1}^{2}}+\sqrt{g_{2}^{2}+f_{2}^{2}}$
Squaring both sides, we get
$-2 g_{1} g_{2}-2 f_{1} f_{2}=2 \sqrt{\left(g_{1}^{2}+f_{1}^{2}\right)\left(g_{2}^{2}+f_{2}^{2}\right)}$
Again squaring, we get
$\left(g_{1} f_{2}\right)^{2}+\left(g_{2} f_{1}\right)^{2}-2 g_{1} g_{2} f_{1} f_{2}=0 $
$\Rightarrow \left(g_{1} f_{2}-g_{2} f_{1}\right)^{2}=0 $
$\Rightarrow \frac{g_{1}}{g_{2}}=\frac{f_{1}}{f_{2}}$