If $h_{1}, h_{2}, h_{3}$ are lengths of altitudes of $∆$ ABC, and $r$ is the in radius then minimum value of $\frac{h_{1} + r}{h_{1} - r} + \frac{h_{2} + r}{h_{2} - r} + \frac{h_{3} + r}{h_{3} - r}$ is

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Detailed Solution

$\begin{array}{l} h_{1} = \frac{2 Δ}{a} \Rightarrow 1 ly h_{2} = \frac{2 Δ}{b} \Rightarrow h_{3} = \frac{2 Δ}{c} \\ \Rightarrow \frac{h_{1} + r}{h_{1} - r} \times \frac{h_{2} + r}{h_{2} - r} \times \frac{h_{3} + r}{h_{3} - r} \\ = Σ (\frac{\frac{2 Δ}{a} + \frac{Δ}{\frac{s}{a}}}{a} - \frac{Δ}{s}) = Σ (\frac{2 s + a}{2 s - a}) = Σ (\frac{4 s}{2 s - a}) - 3 . \\ But \frac{4 s}{2 s - a} + \frac{4 s}{2 s - b} + \frac{4 s}{2 s - c} \geq \\ \Rightarrow \frac{2 s - a}{4 s} + \frac{2 s - b}{2 s} + \frac{2 s - c}{4 s} \\ ∴ \frac{4 s}{2 s - a} + \frac{4 s}{2 s - b} + \frac{4 s}{2 s - c} \geq 9 \Rightarrow GE \geq 6 \end{array}$