Question

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

Answer 1

Answer: 5

$LHS=12sinx+5cosx\in \left[-\sqrt{12^2+5^2},\sqrt{12^2+5^2}\right]$
i.e. $\left[-13,13\right]$ i.e.
maximum value of $LHS$ is $13$
$RHS=2\left(y^2-4y+4\right)+13$
$=2{\left(y-2\right)}^2+13$
$RHS\ge 13$
Roots of the equation exist if $LHS=RHS=13.$
$RHS=13$ when $y=2$
$LHS=13\Rightarrow 12sinx+5cosx=13$
$\Rightarrow \frac{12}{13}sinx+\frac{5}{13}cosx=1$
$sin\left(x+\alpha \right)=1,$ where $tan\alpha =\frac{5}{12}$
$x+ta{n}^{-1}\frac{5}{12}=\frac{\pi }{2}\Rightarrow x=\frac{\pi }{2}-ta{n}^{-1}\frac{5}{12}$
$\Rightarrow xy=\pi -2ta{n}^{-1}\frac{5}{12}\Rightarrow \frac{5}{12}=tan\left(\frac{\pi }{2}-\frac{xy}{2}\right)=cot\left(\frac{xy}{2}\right)$
$\Rightarrow 12cot\left(\frac{xy}{2}\right)=5$