We check collinearity by comparing the slopes of the three pairs: Slope of PR
mPR=
yP−yR
xP−xR
=
sinβ−0
cosα−0
=
sinβ
cosα
>0(. since 0<β<
π
4
)
. Slope of RQ
mRQ=
yQ−yR
xQ−xR
=
cosβ−0
sinα−0
=
cosβ
sinα
>0( since 0<α<
π
4
).
Slope of PQ
mPQ=
cosβ−sinβ
sinα−cosα
=
cosβ−sinβ
−(cosα−sinα)
=−
cosβ−sinβ
cosα−sinα
<0
because in (0,
π
4
) we have cost>sint, so both cosβ−sinβ and cosα−sinα are positive. Since two of the slopes are positive and one is negative, no two of the three points lie on the same straight line. Hence Option D: P,Q,R are non-collinear.