Question

The acute angle included between the lines \[ x\sin\theta - y\cos\theta = 5 \] and \[ x\sin\alpha - y\cos\alpha + 11 = 0 \] is

• A. \( |\theta-\alpha| \)
• B. \( \dfrac{\pi}{4} \)
• C. \( \dfrac{\pi}{3} \)
• D. \( \theta+\alpha \)

Hint:

If the slopes are \( \tan\theta \) and \( \tan\alpha \), the angle between the lines is simply \( |\theta-\alpha| \).

Answer 1

The Correct Option is A

Step 1: Find slopes of the given lines.
From \( x\sin\theta - y\cos\theta = 5 \): \[ y = x\tan\theta - 5\sec\theta \Rightarrow m_1 = \tan\theta \] From \( x\sin\alpha - y\cos\alpha + 11 = 0 \): \[ y = x\tan\alpha + 11\sec\alpha \Rightarrow m_2 = \tan\alpha \]

Step 2: Use the formula for angle between two lines.
\[ \tan\phi = \left|\frac{m_1-m_2}{1+m_1m_2}\right| = \left|\frac{\tan\theta-\tan\alpha}{1+\tan\theta\tan\alpha}\right| \]

Step 3: Apply tangent subtraction identity.
\[ \tan(\theta-\alpha) = \frac{\tan\theta-\tan\alpha}{1+\tan\theta\tan\alpha} \] \[ \Rightarrow \phi = |\theta-\alpha| \]

Step 4: Conclusion.
\[ \boxed{|\theta-\alpha|} \]