If in a triangle ABC, $r_{1} = 2, r_{2} = 3$ and $r_{3} = 6$ then $a =$

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Solution

The correct option is D 3
$r_{1} = 2, r_{2} = 3, r_{3} = 6$
$r_{1} r_{2} + r_{2} r_{3} + r_{3} r_{1} = s^{2}$ ...(1)
subsituting
$\Rightarrow 2.3 + 3.6 + 6.2 = s^{2}$
$\Rightarrow s^{2} = 36$
$\to s = 6$
Also, $\frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = \frac{1}{r}$ ...(2)
$\to \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{1}{r}$
$\Rightarrow \frac{3 + 2 + 1}{6} = \frac{1}{r} \to r = 1$
$\Rightarrow r = \frac{Δ}{s} \Rightarrow r \times s$
$\Rightarrow Δ = 6 \times 1 = 6$
Also, $r_{1} = \frac{Δ}{s - a}$
$\Rightarrow 2 = \frac{6}{6 - a}$
$\Rightarrow 12 - 2 a = 6$
$\Rightarrow 6 = 2 a$
$\Rightarrow a = 3$

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