Question

If $x$ and $y$ satisfy the equation $12\sin x+5\cos x=2y^2-8y+21$, then the value of $12\cot \left(\frac{xy}{2}\right)$ is

Answer 1

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

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

Now, $12\cot \left(\frac{xy}{2}\right)$
$=12\cot x=12\cot (\frac{(4n+1)\pi }{2}-\alpha )=12\tan \alpha =5$