Question

If the equation $x^{2}+4 x \sin \theta+\tan \theta=0$ $\left(0 < \theta < \frac{\pi}{2}\right)$ has repeated roots, then $\theta$ equals

• A. $\frac{\pi}{12}$
• B. $\frac{\pi}{6}$
• C. $\frac{\pi}{12}$ or $\frac{5 \pi}{12}$
• D. $\frac{\pi}{6}$ or $\frac{\pi}{12}$

Answer 1

Answer: (C)

$D =0 \Rightarrow 16 \sin ^{2} \theta=4 \tan \theta$
$\Rightarrow 4 \sin ^{2} \theta=\tan \theta $
$\Rightarrow 2 \sin 2 \theta=1$
$\sin 2 \theta=\frac{1}{2} $
$\Rightarrow 2 \theta=\frac{\pi}{6}$ or $\frac{5 \pi}{6}$
$\Rightarrow \theta=\frac{\pi}{12} $ or $\frac{5 \pi}{12}$