Question

The set of all possible values of $\theta$ in the interval $(0,\pi )$ for which the points $(1,2)$ and $(\sin \theta ,$ $\cos \theta$ ) lie on the same side of the line $x+y=1$ is :

• A. $\left(0,\frac{\pi }{4}\right)$
• B. $\left(0,\frac{3\pi }{4}\right)$
• C. $\left(\frac{\pi }{4},\frac{3\pi }{4}\right)$
• D. $\left(0,\frac{\pi }{2}\right)$

Answer 1

Answer: (D)

Given that both points $(1,2)\&(\sin \theta ,\cos \theta )$ lie
on same side of the line $x+y-1=0$

$So\left(\begin{array}{c}put\left(1,2\right)in\\ givenline\end{array}\right)\left(\begin{array}{c}putsin\theta cos\theta in\\ givenline\end{array}\right)>0$
$\Rightarrow (1+2-1)(\sin \theta +\cos \theta -1)>0$
$\Rightarrow \sin \theta +\cos \theta >1\{÷by\sqrt{2}\}$
$\Rightarrow \frac{1}{\sqrt{2}}\sin \theta +\frac{1}{\sqrt{2}}\cos \theta >\frac{1}{\sqrt{2}}$
$\Rightarrow sin\left(\theta +\frac{\pi }{4}\right)>\frac{1}{\sqrt{2}}$
$\Rightarrow \frac{\pi }{4}<\theta +\frac{\pi }{4}<\frac{3\pi }{4}$
$\Rightarrow 0<\theta <\frac{\pi }{2}$