If $x$ and $y$ satisfies 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

Given : $12 sin x + 5 cos x = 2 y^{2} - 8 y + 21$
$\Rightarrow 13 sin (x + α) = 2 (y - 2)^{2} + 13$
Where $sin α = \frac{5}{13}$

We know that,
$13 sin (x + α) \in [- 13, 13]$
And
$2 (y - 2)^{2} + 13 \geq 13$
So, solution is possible only when
$y = 2 sin (x + α) = 1 \Rightarrow x + α = \frac{(4 n + 1) π}{2}$

Now,
$12 cot (\frac{x y}{2}) = 12 cot x = 12 cot (\frac{(4 n + 1) π}{2} - α) = 12 tan α = 5$

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