Question

The number of values of $\theta$ satisfying $4\cos \theta +3\sin \theta =5$ as well as $3\cos \theta +4\sin \theta =5$ is

• A. $1$
• B. $2$
• C. $0$
• D. None of these

Answer 1

The correct option is C $0$

$4\cos \theta +3\sin \theta =5$ and $3\cos \theta +4\sin \theta =5$
$\Rightarrow \frac{4}{5}\cos \theta +\frac{3}{5}\sin \theta =1$
Let $\sin A=\frac{4}{5}\therefore \cos A=\frac{3}{5}$
$\therefore \sin A\cos \theta +\cos A\sin \theta =1$
$\Rightarrow \sin (A+\theta )=1$
$\Rightarrow A+\theta =\frac{\pi }{2}.....\left(1\right)$
Now second equation $3\cos \theta +4\sin \theta =5$
$\frac{3}{5}\cos \theta +\frac{4}{5}\sin \theta =1$
$\Rightarrow \cos A\cos \theta +\sin A\sin \theta =1$
$\Rightarrow \cos (\theta -A)=1$
$\Rightarrow \theta -A=0$
$\Rightarrow \theta =A$
$\therefore 2\theta =\frac{\pi }{2}$ ........ From $\left(1\right)$
$\Rightarrow \theta =\frac{\pi }{4}$
Put value of $\theta$ in $4\cos \theta +3\sin \theta =5$
LHS= $4\cos \frac{\pi }{4}+3\sin \frac{\pi }{4}=\frac{4}{\sqrt{2}}+\frac{3}{\sqrt{2}}=\frac{7}{\sqrt{2}}\ne 5$
$\therefore$ No solution exists
No. of solutions $=0$