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Entropy Change in Isothermal Process

The exradii o...

Question

The exradii of a triangle, $r_{1}, r_{2}, r_{3}$ are in harmonic progression, then the sides a, b and c are in.

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None of these

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Solution

The correct option is C AP
It is given that, the exradii $r_{1}, r_{2}$ and $r_{3}$ with corresponding sides $a, b, c$ are in HP.
$\Rightarrow \frac{2}{r_{2}} = \frac{1}{r_{1}} + \frac{1}{r_{3}}$
$\Rightarrow \frac{2 (s - b)}{Δ} = \frac{s - a}{Δ} + \frac{s - c}{Δ}$
$\Rightarrow 2 s - 2 b = 2 s - a - c$
$\Rightarrow 2 b = a + c$
Hence, a, b and c are in AP.

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