Three circles are such that each touch the other two externally. The common tangents are concurrent at P. The length of the tangent to each circle is p. The ratio of the product of their radii to sum of their radii is

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$\frac{p^{2}}{2}$

$p$

$p^{2}$

$\frac{p}{2}$

answer is C.

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Detailed Solution

P is the incentre of $∆$ ^leABC
$\tan (\frac{A}{2}) = \frac{p}{r_{1}}$
Since in $Δ^{ℓe} ABC, \sum \cot \frac{A}{2} = πcot \frac{A}{2}$
$ie. \frac{r_{1} r_{2} r_{3}}{r_{1} + r_{2} + r_{3}} = p$

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