Question

$P(\sin \theta ,\cos \theta )$ and $Q(\cos \theta ,\sin \theta )$ are two points and line joining them has mid point at the origin. $R(\sin 2\theta ,\cos \theta )$ is a point on the plane whose distance from the origin is

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

Answer 1

The correct option is C $\sqrt{\frac{3}{2}}$

from the origin as the midpoint, we get

$\sin \theta +\cos \theta =0$ .........(1)

and by squaring it, we get

$\sin 2\theta =-1$....(2)

from (1) and (2), we get $\cos \theta =\pm \frac{1}{\sqrt{2}}$,Now coordinates of R become $(-1,\cos \theta )$,it's distance from origin will be

$\sqrt{1+\frac{1}{2}}=\sqrt{\frac{3}{2}}$