Consider two circles in a plane with radii 5 and 3 respectively. Distance between centres of these two circles is 10. If $R_{1}$ is radius of circle passing through 4 points of contact of direct common tangents of these two circles and $R_{2}$ is radius of circle passing through 4 points of contact of transverse common tangents then $\frac{R_{1}}{R_{2}} = ?$

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$\frac{5}{3}$

$\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$

$\frac{25}{9}$

answer is B.

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Detailed Solution

$F G = \frac{r_{1} + r_{2}}{2}$

$G B = \sqrt{{(F G)}^{2} + {(F B)}^{2}}$

${(G B)}^{2} = \frac{{(r_{1} + r_{2})}^{2}}{4} + \frac{d^{2} - {(r_{1} - r_{2})}^{2}}{4} = \frac{d^{2} + 4 r_{1} r_{2}}{4}$

$R_{1}^{2} = \frac{{(C_{1} C_{2})}^{2} + 4 r_{1} r_{1}}{4} = 40, R_{2}^{2} = \frac{{(C_{1} C_{2})}^{2} - 4 r_{1} r_{1}}{4} = 10$

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