Question

The line $L$ is passing through $(1, 2, 3)$. The distance of any point on the line $L$ from the line $\vec{r} = (3\lambda - 1)\hat{i} + (-2\lambda + 3)\hat{j} + (4 + \lambda)\hat{k}$ is constant. Then the line $L$ does not pass through the point ______. 

• A. $(4, 0, 4)$
• B. $(-2, 4, 2)$
• C. $(7, -2, 5)$
• D. $(-5, 6, 2)$

Hint:

Constant distance between two lines implies the lines are parallel.

Answer 1

The Correct Option is D

Step 1: Concept
If the distance of every point on line $L$ from a fixed line $M$ is constant, line $L$ must be parallel to line $M$.
Step 2: Analysis
The direction ratios of the given line are $(3, -2, 1)$.
Line $L$ passes through $(1, 2, 3)$ and has direction ratios $(3, -2, 1)$.
Equation of $L$: $\frac{x-1}{3} = \frac{y-2}{-2} = \frac{z-3}{1}$.
Step 3: Verification
Check if $(-5, 6, 2)$ lies on $L$:
$\frac{-5-1}{3} = -2$; $\frac{6-2}{-2} = -2$; $\frac{2-3}{1} = -1$.
Since $-2 \neq -1$, the point $(-5, 6, 2)$ does not lie on the line.
Final Answer: (D)