Question

If $x$ and $y$ are the solutions of the equation $12\sin x+5\cos x=2y^2-8y+21$, the value of $3864\cot \left(\frac{xy}{2}\right)$ is

Answer 1

Given equation $12\sin x+5\cos x=2y^2-8y+21$
Consider LHS =$12\sin x+5\cos x$
Put $12=r\cos \alpha ,5=r\sin \alpha$
Squaring and adding , we get
$\Rightarrow r=13,\tan \alpha =\frac{5}{12}$
So LHS =$13\sin (x+\alpha )$
Since, $\sin (x+\alpha )\le 1$
$\Rightarrow 13\sin (x+\alpha )\le 13$
So, $LHS\le 13$
Now, consider $RHS=2y^2-8y+21$
$=2(y-2{)}^2+13$
Since $2(y-2{)}^2\ge 0$
$\Rightarrow 2(y-2{)}^2+13\ge 13$
So, $RHS\ge 13$
Hence, roots of the equation exist if L.H.S. $=$ R.H.S. $=$ 13
$\Rightarrow y=2$ and $\sin (x+{\tan }^{-1}(\frac{5}{12}\left)\right)=1$
$\Rightarrow x=\frac{\pi }{2}-{\tan }^{-1}\frac{5}{12}$
$\Rightarrow x={\cot }^{-1}\frac{5}{12}$
$\therefore 3864\cot \left(\frac{xy}{2}\right)=3864\times \frac{5}{12}$
$=322\times 5=1610$