Straight Line and Pair of Straight Lines 2 Question 2

2. 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 $0<\alpha, \beta, \theta<\frac{\pi}{4}$. Then,

$(2008,4 M)$

(a) $P$ lies on the line segment $R Q$

(b) $Q$ lies on the line segment $P R$

(c) $R$ lies on the line segment $Q P$

(d) $P, Q, R$ are non-colinear

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Answer:

Correct Answer: 2. (d)

Solution:

  1. For collinear points

$$ \Delta=\left|\begin{array}{ccc} -\sin (\beta-\alpha) & -\cos \beta & 1 \\ \cos (\beta-\alpha) & \sin \beta & 1 \\ \cos (\beta-\alpha+\theta) & \sin (\beta-\theta) & 1 \end{array}\right| $$

Clearly, $\Delta \neq 0$ for any value of $\alpha, \beta, \theta$.

Hence, points are non-collinear.



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