The range of values of \( \theta \) in the interval \( \left( 0, \pi \right) \) such that the points \( (3, 2) \) and \( (\cos \theta, \sin \theta) \) lie on the same sides of the line \( x + y - 1 = 0 \), is:
Show Hint
- \( \left( 0, \frac{3\pi}{4} \right) \)
- \( \left( 0, \frac{\pi}{2} \right) \)
- \( \left( 0, \frac{\pi}{3} \right) \)
- \( \left( 0, \frac{\pi}{4} \right) \)
The Correct Option is B
Solution and Explanation
Step 1: Understand the equation of the line.
The equation of the line is \( x + y - 1 = 0 \), which can be rewritten as:
\[
x + y = 1.
\]
This is a straight line with slope \( -1 \) passing through the point \( (1, 0) \) on the \( x \)-axis and the point \( (0, 1) \) on the \( y \)-axis. The points \( (3, 2) \) and \( (\cos \theta, \sin \theta) \) must lie on the same side of this line.
Step 2: Check the side of the line for point \( (3, 2) \).
To determine which side of the line the point \( (3, 2) \) lies on, substitute \( x = 3 \) and \( y = 2 \) into the equation of the line:
\[
3 + 2 = 5.
\]
Since \( 5 > 1 \), the point \( (3, 2) \) lies on the side of the line where \( x + y > 1 \).
Step 3: Check the side of the line for the point \( (\cos \theta, \sin \theta) \).
For the point \( (\cos \theta, \sin \theta) \), substitute \( x = \cos \theta \) and \( y = \sin \theta \) into the equation of the line:
\[
\cos \theta + \sin \theta.
\]
For the points to lie on the same side of the line, we need \( \cos \theta + \sin \theta > 1 \). Now, we will find the range of \( \theta \) for which this inequality holds.
Step 4: Solve the inequality.
To solve \( \cos \theta + \sin \theta > 1 \), square both sides:
\[
(\cos \theta + \sin \theta)^2 > 1^2,
\]
\[
\cos^2 \theta + 2\cos \theta \sin \theta + \sin^2 \theta > 1,
\]
\[
1 + 2\cos \theta \sin \theta > 1.
\]
This simplifies to:
\[
2\cos \theta \sin \theta > 0.
\]
Since \( \cos \theta \sin \theta = \frac{1}{2} \sin(2\theta) \), the inequality becomes:
\[
\sin(2\theta) > 0.
\]
The sine function is positive in the interval \( \left( 0, \frac{\pi}{2} \right) \), so the condition holds for \( \theta \in \left( 0, \frac{\pi}{2} \right) \).
Step 5: Conclusion.
Thus, the range of \( \theta \) is \( \left( 0, \frac{\pi}{2} \right) \), and the correct answer is (b).
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