First slide

Cartesian plane

Question

Consider three points $P (\cos α, \sin β),$ $Q (\sin α, \cos β)$ and $R (0, 0) .$ Where $0 < α, β < π / 4$ Then

Moderate

A

$P$ lies on the line segment $R Q$
B

$Q$ lies on the line segment $P Q$
C

$R$ lies on the line segment $Q P$
D

$P, Q, R$ are non-collinear.

Solution

Consider $Δ = |\begin{array}{ccc} \cos α & \sin β & 1 \\ \sin α & \cos β & 1 \\ 0 & 0 & 1 \end{array}|$
$= \cos α \cos β - \sin α \sin β$
$= \cos (α + β) \neq 0$ as $0 < α + β < π / 2$
Hence the given points are non-collinear.

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