Question

If $\tan \theta ,2\tan \theta +2,3\tan \theta +3$ are in G.P., then the value of $\frac{7-5\cot \theta }{9-4\sqrt{{\sec }^2\theta -1}}$ is

• A. $\frac{12}{5}$
• B. $-\frac{33}{28}$
• C. $\frac{33}{100}$
• D. $\frac{12}{13}$

Answer 1

The correct option is B $-\frac{33}{28}$

According to the given condition, $(2\tan \theta +2{)}^2=(3\tan \theta +3)\tan \theta$
$\Rightarrow 4{\tan }^2\theta +8\tan \theta +4=3{\tan }^2\theta +3\tan \theta$
$\Rightarrow {\tan }^2\theta +5\tan \theta +4=0$
So, $\tan \theta =-1$ or $-4$, but $-1$ doesn't satisfy the G.P. condition as two terms become $0$.
$\frac{7-5\cot \theta }{9-4\sqrt{{\sec }^2\theta -1}}=\frac{7+1.25}{9-16}=\frac{8.25}{-7}=\frac{-33}{28}$