If $x$ $(| x | < π)$ and $y$ are the solutions of the equation $12 sin x + 5 cos x = 2 y^{2} - 8 y + 21,$ then the value of $12 cot (\frac{x y}{2})$ is

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Solution

$- \sqrt{12^{2} + 5^{2}} \leq 12 sin x + 5 cos x \leq \sqrt{12^{2} + 5^{2}}$
i.e., $12 sin x + 5 cos x \in [- 13, 13]$

RHS $= 2 (y^{2} - 4 y + 4) + 13 = 2 (y - 2)^{2} + 13$
RHS $\geq 13$ and LHS $\leq 13$
So, the equality holds only when LHS $=$ RHS $= 13$

RHS $= 13$ when $y = 2$
LHS $= 13$ when $\frac{12}{13} sin x + \frac{5}{13} cos x = 1$
$\Rightarrow sin (x + α) = 1$ where $tan α = \frac{5}{12}$
$\Rightarrow x = \frac{π}{2} - α$

$∴ \frac{x y}{2} = (\frac{π}{2} - α)$
$\Rightarrow cot (\frac{x y}{2}) = cot (\frac{π}{2} - α) = tan α = \frac{5}{12}$
$\Rightarrow 12 cot (\frac{x y}{2}) = 5$

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