Question

If the lines \(\frac{x-5}{5m+2} = \frac{2-y}{5} = \frac{1-z}{-1}\) and \(x = \frac{2y+1}{4m} = \frac{1-z}{-3}\) are perpendicular to each other, then the value of m is ...

• A. \(1\)
• B. \(0\)
• C. \(2\)
• D. \(-1\)

Hint:

Get direction ratios (5m+2, -5, 1) and (1, 2m, 3).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Two lines are perpendicular when the dot product of their direction ratios is zero. First write each line in the form \(\dfrac{x-x_1}{l}=\dfrac{y-y_1}{m}=\dfrac{z-z_1}{n}\).

Step 2: Line 1:
\(\dfrac{x-5}{5m+2}=\dfrac{2-y}5=\dfrac{1-z}{-1}\) can be written as \(\dfrac{x-5}{5m+2}=\dfrac{y-2}{-5}=\dfrac{z-1}{1}\). Direction ratios: \((5m+2,\,-5,\,1)\).

Step 3: Line 2:
\(x=\dfrac{2y+1}{4m}=\dfrac{1-z}{-3}\) can be written as \(\dfrac{x}{1}=\dfrac{y+\frac12}{2m}=\dfrac{z-1}{3}\). Direction ratios: \((1,\,2m,\,3)\).

Step 4: Perpendicular condition:
\[ (5m+2)(1)+(-5)(2m)+(1)(3)=0 \]
\[ 5m+2-10m+3=0\ \Rightarrow\ 5-5m=0\ \Rightarrow\ m=1 \]

Step 5: Choose:
Option (A).

Final Answer:
m = 1. \[ \boxed{1} \]