Consider a hexagon incribed in a circle of radius $' r'$ . If the hexagon has two sides of length $2$ , two sides of length $7$ and two sides of length $11$ , then $h^{3} - m r - n = 0$ where

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$n = 77$

$m = 87$

$(1, m, n) = (2, - 87, - 77)$

$I = 2$

answer is A, B, C.

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Detailed Solution

Here $A D$ is a diameter of the circumcircle of the hexagon and $r$ be the radius of the circle.

$∴ B D = \sqrt{4 r^{2} - 4} and A C = \sqrt{4 r^{2} - 121}$

and $A B C D$ is a cyclic quadrailateral.

$\Rightarrow AB \cdot CD + AD \cdot BC = AC \cdot BD$

$\begin{array}{l} \Rightarrow 2.11 + 2 r .7 = \sqrt{4 r^{2} - 4} \sqrt{4 r^{2} - 121} \\ \Rightarrow (r^{2} - 1) (4 r^{2} - 121) = (11 + 7 r)^{2} \Rightarrow 4 {(r^{2})}^{2} \\ - 125 r^{2} + 121 = 121 + 49 r^{2} + 154 r \end{array}$

$\begin{array}{l} \Rightarrow 4 {(r^{2})}^{2} - 174 r^{2} - 154 r = 0 \Rightarrow 4 r^{3} - 174 r - 154 = 0 \\ \Rightarrow 2 r^{3} - 87 r - 77 = 0 \end{array}$