If the two circles, $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0$ and $x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y = 0$ touches each other, then-

f_{1} g_{1} = f_{2} g_{2}

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\frac{f_{1}}{g_{1}} = \frac{f_{2}}{g_{2}}

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f_{1} f_{2} = g_{1} g_{2}

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Solution

The correct option is B $\frac{f_{1}}{g_{1}} = \frac{f_{2}}{g_{2}}$
$C_{1} C_{2} = r_{1} \pm r_{2}$
$\Rightarrow$ ${(g_{1} - g_{2})}^{2} + {(f_{1} - f_{2})}^{2} = {(\sqrt{g_{1}^{2} + f_{1}^{2}} \pm \sqrt{g_{2}^{2} + f_{2}^{2}})}^{2}$
$\Rightarrow$ $- 2 g_{1} g_{2} - 2 f_{1} f_{2} = \pm 2 {(\sqrt{g_{1}^{2} + f_{1}^{2}} . \sqrt{g_{2}^{2} + f_{2}^{2}})}^{2}$
$\Rightarrow$ $g_{1} f_{2} - g_{2} f_{1} = 0$
$\Rightarrow$ $\frac{g_{1}}{g_{2}} = \frac{f_{1}}{f_{2}}$
Ans: B

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x^{2} + y^{2} + 2 g x + 2 f y = 0

x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0

If the two circles $x^{2} + y^{2} + 2 g x + 2 f y = 0$ and $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0$ touch each other then

x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0 & x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y = 0

x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y = 0

x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y = 0

f_{1} g_{2} = f_{2} g_{1}

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