Question

Let the point $P(2\cos \theta ,2\sin \theta ),$ where $\theta ϵ(-\pi ,\pi ),$ does not fall in the angle between the lines $y=x-2$ and $y=2-x$ in which the origin lies. If the least and greatest values taken by $\theta$ are A and B, respectively, then $\sin A+\cos B$ =

• A. $1$
• B. $-1$
• C. $\sqrt{2}$
• D. $\frac{\sqrt{3}+1}{2}$

Answer 1

Answer: (B)

$-1$

Given lines

$y=x-2$ and $y=2-x$

Equation of angle bisector

$y-x+2=\pm (y-2+x)$

$y-x+2=y-2+x$ and $y-x+2=-y+2-x$

$x=2$ and $y=0$

Since the Point $P(2\cos \theta ,2\sin \theta )$ does not lies in between the angle bisector

$2\cos \theta -2>0$

$\cos \theta -1>0$

$\cos \theta >1\Rightarrow \theta >0$

Hence the range of $\theta ϵ(-\pi ,\pi )$

But $\theta >0$

So least value of $\theta =0=A$ and Greatest value $B=\theta =\pi$

$\Rightarrow \sin A+\cos B=\sin 0+\cos \pi =0-1=-1$