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Question Id : 11544 | Context :NIMCET 2023

Question

The range of values of $\theta$ in the interval $(0,\pi)$ such that the points (3, 2) and $(cos\theta ,sin\theta)$ lie on the samesides of the line x + y – 1 = 0, is
🎥 Video solution / Text Solution of this question is given below:

Same Side of a Line — Geometric Condition

Line: \( x + y - 1 = 0 \)
Point 1: (3, 2) → Lies on the side where value is positive:
\[ f(3, 2) = 3 + 2 - 1 = 4 > 0 \]

Point 2: \( (\cos\theta, \sin\theta) \) lies on same side if: \[ \cos\theta + \sin\theta > 1 \] Using identity: \[ \cos\theta + \sin\theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \Rightarrow \sin\left(\theta + \frac{\pi}{4}\right) > \frac{1}{\sqrt{2}} \] So: \[ \theta \in \left(0,\ \frac{\pi}{2}\right) \]

✅ Final Answer: \( \boxed{\theta \in \left(0,\ \frac{\pi}{2}\right)} \)

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