First slide

Cartesian plane

Question

Consider three points
$\begin{array}{l} P \equiv (- \sin (β - α), - cosβ), \\ Q \equiv (\cos (β - α), sinβ), and R \equiv (\cos (β - α + θ), \sin (β - θ)), where 0 < α, β, θ < \frac{π}{4} . Then \end{array}$

Moderate

A

P lies on the line segment RQ
B

Q lies on the line segment PR
C

R lies on the line segment QP
D

P,R,R are non -collinear

Solution

$P \equiv (- \sin (β - α), - cosβ) \equiv (x_{1}, y_{1})$
$Q \equiv (\cos (β - α), sinβ) \equiv (x_{2}, y_{2})$
$R \equiv (\cos (β - α + θ), \sin (β - θ))$
or
$R \equiv (x_{2} cosθ + x_{1} sinθ, y_{2} cosθ + y_{1} sinθ)$
If $T \equiv (\frac{x_{2} cosθ + x_{1} sinθ}{cosθ + sinθ}, \frac{y_{2} cosθ + y_{1} sinθ}{cosθ + sinθ})$ , for some $' θ'$ then P,Q and T are collinear.
$∴$ P,Q,R are collinear if $cosθ + sinθ = 1,$ which is not possible as $0 < θ < \frac{π}{4} .$
Hence, P,Q,R are non-collinear.

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