Question

The range of values of $\theta$ in the interval $(0,\pi )$ such that the points $(3,2)$ and $(\cos \theta ,\sin \theta )$ lie on the same side of the line $x+y-1=0$, is

• A. $(0,\frac{\pi }{2})$
• B. $(0,\frac{\pi }{4})$
• C. $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$
• D. None of these

Answer 1

The correct option is A $(0,\frac{\pi }{2})$

Since, the points $(3,2)$ and $(\cos \theta ,\sin \theta )$ lie on the same side of the line $x+y-1=0$, then
$3+2-1$ and $\cos \theta +\sin \theta -1$ must be of the same sign.
$\therefore \cos \theta +\sin \theta -1>0$
$\Rightarrow \frac{1}{\sqrt{2}}\cos \theta +\frac{1}{\sqrt{2}}\sin \theta >\frac{1}{\sqrt{2}}$
$\cos \theta \sin \frac{\pi }{4}+\sin \theta \cos \frac{\pi }{4}>\frac{1}{\sqrt{2}}$
$\Rightarrow \sin (\theta +\frac{\pi }{4})>\frac{1}{\sqrt{2}}$
Since, $0<\theta <\pi$
$\Rightarrow \frac{\pi }{4}<\theta +\frac{\pi }{4}<\frac{3\pi }{4}$
$\Rightarrow 0<\theta <\frac{\pi }{2}$
$\Rightarrow \theta \in (0,\frac{\pi }{2})$