Consider three points `P = (-sin (beta-alpha), -cos beta)`, `Q = (cos(beta-alpha), sin beta)`, and `R = ((cos (beta - alpha + theta), sin (beta - theta))`, where `0< alpha, beta, theta < pi/4` Then

Options

A. P lies on the line segment RQ
B. Q lies on the segment PR
C. R lies on the line segment PR
D. P,Q,R are non-collinear

Answer

D

Explanation

`P-=(-sin(beta-alpha),-cosbeta)-=(x_1,y_1)`
`Q-=(cos(beta-alpha),sinbeta)-=(x_2,y_2)`
`R-=(cos(beta-alpha+theta),sin(beta-theta))`
or `R-=(x_2costheta+x_1sintheta,y_2costheta+y_1sintheta)`
if `T-=((x_2+costheta+x_1sintheta)/(costheta+sintheta),(y_2+costheta+y_1sintheta)/(costheta+sintheta))`,for some `theta` then P,Q, and T are collinear.
`therefore` P,Q,R are collinear if `costheta+sintheta=1`, which is not possible as `0ltthetaltpi//4`.
Hence, P,Q,R are non -collinear.