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:
- 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.
