Question

A line \( L \) passing through the point \( P(-5,-4) \) cuts the lines \( x - y - 5 = 0 \) and \( x + 3y + 2 = 0 \) respectively at \( Q \) and \( R \) such that \[ \frac{18}{PQ} + \frac{15}{PR} = 2, \] then the slope of the line \( L \) is:

• A. \( \pm 1 \)
• B. \( \pm \frac{1}{\sqrt{3}} \)
• C. \( \pm \sqrt{3} \)
• D. \( \pm \frac{2}{\sqrt{3}} \)

Hint:

To solve problems involving lines and given conditions on distances, use the section formula and distance properties in coordinate geometry.

Answer 1

The Correct Option is C

Step 1: Understanding given equation 
The given equation states that the sum of reciprocals of distances \( PQ \) and \( PR \) is given by: \[ \frac{18}{PQ} + \frac{15}{PR} = 2. \] This equation can be transformed using the section formula in coordinate geometry. 

Step 2: Finding intersection points 
The given lines are: \[ x - y - 5 = 0 \quad \Rightarrow \quad y = x - 5. \] \[ x + 3y + 2 = 0 \quad \Rightarrow \quad y = -\frac{x}{3} - \frac{2}{3}. \] The parametric equation of a line passing through \( P(-5,-4) \) with slope \( m \) is: \[ y + 4 = m (x + 5). \] Solving for \( Q \) and \( R \) using this line equation, we find the required distances \( PQ \) and \( PR \). 

Step 3: Solving for slope \( m \) 
After substituting in the distance equation and solving, we obtain: \[ m = \pm \sqrt{3}. \] 

Step 4: Conclusion 
Thus, the final answer is: \[ \boxed{\pm \sqrt{3}}. \]