Question

If $A(\sin \alpha ,\frac{1}{\sqrt{2}})$ and $B(\frac{1}{\sqrt{2}},\cos \alpha ),-\pi \le \alpha \le \pi$, are two points on the same side of the line $x-y=0$ then $\alpha$ belongs to the interval

• A. $(-\frac{\pi }{4},\frac{\pi }{4})\cup (\frac{\pi }{4},\frac{3\pi }{4})$
• B. $(-\frac{\pi }{4},\frac{\pi }{4})$
• C. $(\frac{\pi }{4},\frac{3\pi }{4})$
• D. none of these

Answer 1

The correct option is A $(-\frac{\pi }{4},\frac{\pi }{4})\cup (\frac{\pi }{4},\frac{3\pi }{4})$

For point A and B lie on same side of line

$\sin \alpha -\frac{1}{\sqrt{2}}>0$ or $\sin \alpha -\frac{1}{\sqrt{2}}<0$ ...(1)
And

$\frac{1}{\sqrt{2}}-\cos \alpha >0$ or $\frac{1}{\sqrt{2}}-\cos \alpha <0$ ...(2)
Solving (1) and (2) we get

$\alpha ϵ(-\frac{\pi }{4},\frac{\pi }{4})\cup (\frac{\pi }{4},\frac{3\pi }{4})$