Question

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

• A. $8$
• B. $12$
• C. $5$
• D. $1$

Answer 1

The correct option is C $5$

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

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

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

$\therefore \frac{xy}{2}=(\frac{\pi }{2}-\alpha )$
$\Rightarrow \cot \left(\frac{xy}{2}\right)=\cot (\frac{\pi }{2}-\alpha )=\tan \alpha =\frac{5}{12}$
$\Rightarrow 12\cot \left(\frac{xy}{2}\right)=5$