Question

Let \( A \) be the point of intersection of the lines \( 3x + 2y = 14 \), \( 5x - y = 6 \) and \( B \) be the point of intersection of the lines \( 4x + 3y = 8 \), \( 6x + y = 5 \). The distance of the point \( P(5, -2) \) from the line \( AB \) is

• A. \( \frac{13}{2} \)
• B. 8
• C. \( \frac{5}{2} \)
• D. 6

Answer 1

The Correct Option is D

Step 1. Find the coordinates of \( A \) by solving the lines \( L_1: 3x + 2y = 14 \) and \( L_2: 5x - y = 6 \):  
  Solving these equations gives \( A(2, 4) \).

Step 2. Find the coordinates of \( B \) by solving the lines \( L_3: 4x + 3y = 8 \) and \( L_4: 6x + y = 5 \):  
  Solving these equations gives \( B\left(\frac{1}{2}, 2\right) \).

Step 3. Determine the equation of line \( AB \) passing through points \( A(2, 4) \) and \( B\left(\frac{1}{2}, 2\right) \):
  The equation of \( AB \) is \( 4x - 3y + 4 = 0 \).

Step 4. Calculate the distance from \( P(5, -2) \) to the line \( AB: 4x - 3y + 4 = 0 \):  
 
  \(\text{Distance} = \frac{|4(5) - 3(-2) + 4|}{\sqrt{4^2 + (-3)^2}} = \frac{|20 + 6 + 4|}{\sqrt{16 + 9}} = \frac{30}{5} = 6.\)
  

So, the distance of point \( P \) from the line \( AB \) is 6.

The Correct Answer is: 6