Question

A straight line through the point P(1,2) makes an angle $\theta$ with the positive X-axis in anti-clockwise direction and meets the line $x+\sqrt{3}y-2\sqrt{3}=0$ at Q. If $PQ = \frac{1}{2}$, then $\theta=$

• A. $\frac{\pi}{6}$
• B. $\frac{5\pi}{6}$
• C. $\frac{2\pi}{3}$
• D. $\frac{\pi}{3}$

Hint:

In problems involving distance along a line in parametric form, remember that the distance $d$ corresponds to a directed distance $r = \pm d$. You may need to check both cases, as one might lead to a valid solution within the given options while the other might not.

Answer 1

The Correct Option is D

Step 1: Parametric Form of Line: Equation of line through \( P(1,2) \): \( x = 1 + r\cos\theta \), \( y = 2 + r\sin\theta \). This line meets \( x+\sqrt{3}y-2\sqrt{3}=0 \) at Q, where \( r = PQ = 1/2 \).
Step 2: Substitute into Line Equation: Substitute \( (x,y) \) into the given line equation: \( (1 + r\cos\theta) + \sqrt{3}(2 + r\sin\theta) - 2\sqrt{3} = 0 \) \( 1 + r\cos\theta + 2\sqrt{3} + \sqrt{3}r\sin\theta - 2\sqrt{3} = 0 \) \( 1 + r(\cos\theta + \sqrt{3}\sin\theta) = 0 \) \( |r(\cos\theta + \sqrt{3}\sin\theta)| = 1 \)
Step 3: Solve for \( \theta \): Given distance \( |r| = 1/2 \). \( \frac{1}{2} |\cos\theta + \sqrt{3}\sin\theta| = 1 \) \( |\cos\theta + \sqrt{3}\sin\theta| = 2 \) Divide by 2: \( |\frac{1}{2}\cos\theta + \frac{\sqrt{3}}{2}\sin\theta| = 1 \) Using \( \cos(A-B) \): \( \cos(\theta - 60^\circ) = \pm 1 \) So \( \theta - 60^\circ = 0 \) (since line usually implies \( \theta \in [0, \pi) \)). \( \theta = 60^\circ = \frac{\pi}{3} \).