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

Answer: (D)

6

Answer 2

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