Question

If the straight line passing through \( P(3,4) \) makes an angle \( \frac{\pi}{6} \) with the positive x-axis in the anticlockwise direction and meets the line \( 12x + 5y + 10 = 0 \) at \( Q \), then the length of the segment \( PQ \) is:

• A. \( \frac{64}{12\sqrt{2} + 1} \)
• B. \( \frac{96}{9\sqrt{2} - 1} \)
• C. \( \frac{112}{10\sqrt{3} + 3} \)
• D. \( \frac{132}{12\sqrt{3} + 5} \)

Hint:

For line intersection problems, find the equation of the given line, substitute into the second equation, solve for \( x, y \), and compute the distance using the distance formula.

Answer 1

The Correct Option is D

To solve the problem, we proceed as follows: Step 1: Find the equation of the straight line passing through \( P(3,4) \) The line makes an angle \( \frac{\pi}{6} \) with the positive x
-axis. The slope \( m \) of the line is: \[ m = \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}. \] Using the point
-slope form, the equation of the line is: \[ y
- 4 = \frac{1}{\sqrt{3}} (x
- 3). \] Multiply through by \( \sqrt{3} \): \[ \sqrt{3} y
- 4\sqrt{3} = x
- 3. \] Rearrange: \[ x
- \sqrt{3} y + 4\sqrt{3}
- 3 = 0. \] Step 2: Find the point of intersection \( Q \) with the line \( 12x + 5y + 10 = 0 \) Solve the system of equations: 1. \( x
- \sqrt{3} y + 4\sqrt{3}
- 3 = 0 \), 2. \( 12x + 5y + 10 = 0 \). From equation 1, express \( x \) in terms of \( y \): \[ x = \sqrt{3} y
- 4\sqrt{3} + 3. \] Substitute into equation 2: \[ 12 (\sqrt{3} y
- 4\sqrt{3} + 3) + 5y + 10 = 0. \] Simplify: \[ 12\sqrt{3} y
- 48\sqrt{3} + 36 + 5y + 10 = 0. \] Combine like terms: \[ (12\sqrt{3} + 5) y
- 48\sqrt{3} + 46 = 0. \] Solve for \( y \): \[ y = \frac{48\sqrt{3}
- 46}{12\sqrt{3} + 5}. \] Substitute back into the expression for \( x \): \[ x = \sqrt{3} \left( \frac{48\sqrt{3}
- 46}{12\sqrt{3} + 5} \right)
- 4\sqrt{3} + 3. \] Simplify: \[ x = \frac{144
- 46\sqrt{3}}{12\sqrt{3} + 5}
- 4\sqrt{3} + 3. \] Step 3: Compute the distance \( PQ \) The distance between \( P(3,4) \) and \( Q(x,y) \) is: \[ PQ = \sqrt{(x
- 3)^2 + (y
- 4)^2}. \] Substitute the expressions for \( x \) and \( y \): \[ PQ = \sqrt{\left( \frac{144
- 46\sqrt{3}}{12\sqrt{3} + 5}
- 4\sqrt{3} + 3
- 3 \right)^2 + \left( \frac{48\sqrt{3}
- 46}{12\sqrt{3} + 5}
- 4 \right)^2}. \] Simplify: \[ PQ = \sqrt{\left( \frac{144
- 46\sqrt{3}
- 4\sqrt{3}(12\sqrt{3} + 5)}{12\sqrt{3} + 5} \right)^2 + \left( \frac{48\sqrt{3}
- 46
- 4(12\sqrt{3} + 5)}{12\sqrt{3} + 5} \right)^2}. \] Further simplification yields: \[ PQ = \frac{132}{12\sqrt{3} + 5}. \] Final Answer: \[ \boxed{\frac{132}{12\sqrt{3} + 5}} \]