Question

If the shortest distance between the lines $\frac{x - k}{2} = \frac{y - 4}{3} = \frac{z - 3}{4}$ and $\frac{x - 2}{4} = \frac{y - 4}{6} = \frac{z - 7}{8}$ is $\frac{13}{\sqrt{29}}$, then $k =$}

• A. $1$
• B. $-1$
• C. $2$
• D. $-2$

Hint:

The two lines are parallel, so use the distance between parallel lines with the cross product of the connecting vector and the direction.

Answer 1

The Correct Option is A

Step 1: Check k = 1
With $k=1$: $\overrightarrow{AB}=(1,0,4)$, $\overrightarrow{AB}\times\vec d=(-12,4,3)$.

Step 2: Magnitude
$|\cdot|=\sqrt{144+16+9}=13$, $|\vec d|=\sqrt{29}$.

Step 3: Distance
$\dfrac{13}{\sqrt{29}}$, as required, so $k=1$.

Final Answer:
k equals 1. \[ \boxed{\text{(A) }1} \]