Question

Find the value of \(k\) if the lines \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) and \( \frac{x-4}{k} = \frac{y-1}{2} = \frac{z}{1} \) are perpendicular.

• A. \(5\)
• B. \(-5\)
• C. \(2\)
• D. \(-2\)

Hint:

If two lines in 3D are perpendicular, the dot product of their direction ratios must be zero.

Answer 1

The Correct Option is B

Concept: Two lines are perpendicular if the dot product of their direction ratios is zero. \[ a_1a_2 + b_1b_2 + c_1c_2 = 0 \] where \( (a_1,b_1,c_1) \) and \( (a_2,b_2,c_2) \) are direction ratios of the lines.

Step 1: Identify the direction ratios of both lines. For the first line: \[ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \] Direction ratios: \[ (2,\,3,\,4) \] For the second line: \[ \frac{x-4}{k} = \frac{y-1}{2} = \frac{z}{1} \] Direction ratios: \[ (k,\,2,\,1) \]

Step 2: Apply the perpendicular condition. \[ 2(k) + 3(2) + 4(1) = 0 \] \[ 2k + 6 + 4 = 0 \]

Step 3: Solve for \(k\). \[ 2k + 10 = 0 \] \[ k = -5 \] \[ \boxed{k = -5} \]