Question

Consider the line $L$ passing through the points $(1, 2, 3)$ and $(2, 3, 5)$. The distance of the point $\left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right)$ from the line $L$ along the line $\frac{3x - 11}{2} = \frac{3y - 11}{1} = \frac{3z - 19}{2}$ is equal to:

• A. 3
• B. 5
• C. 4
• D. 6

Answer 1

Answer: (A)

3

Answer 2

Step 1: Direction ratios of the given line

The direction ratios of the line:

$\frac{3x - 11}{2} = \frac{3y - 11}{1} = \frac{3z - 19}{2}$

are:

$\vec{d} = \langle 2, 1, 2 \rangle$

Step 2: Representing point $B$

Let point $B$ on the given line be:

$B = (1 + \lambda, 2 + \lambda, 3 + 2\lambda),$

where $\lambda$ is the parameter.

Step 3: Direction ratios of line $AB$

Point $A = \left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$. The direction ratios of $AB$ are:

$\text{D.R. of } AB = \left\langle \frac{3\lambda - 8}{3}, \frac{3\lambda - 5}{3}, \frac{6\lambda - 10}{3} \right\rangle$

Step 4: Parallel condition

Since $AB$ lies along the direction vector $\vec{d} = \langle 2, 1, 2 \rangle$, we have:

$\frac{\frac{3\lambda - 8}{3}}{2} = \frac{\frac{3\lambda - 5}{3}}{1} = \frac{\frac{6\lambda - 10}{3}}{2}$

Simplify the first ratio:

$\frac{3\lambda - 8}{3 \cdot 2} = \frac{3\lambda - 5}{3}$

Cross-multiply:

$3\lambda - 8 = 6\lambda - 10$

Solve for $\lambda$:

$3\lambda = 2 \implies \lambda = \frac{2}{3}$

Step 5: Find $B$

Substitute $\lambda = \frac{2}{3}$ into $B = (1 + \lambda, 2 + \lambda, 3 + 2\lambda)$:

$B = \left(1 + \frac{2}{3}, 2 + \frac{2}{3}, 3 + 2 \cdot \frac{2}{3}\right) = \left(\frac{5}{3}, \frac{8}{3}, \frac{13}{3}\right)$

Step 6: Find distance $AB$

The distance $AB$ is given by:

$AB = \sqrt{\left(\frac{11}{3} - \frac{5}{3}\right)^2 + \left(\frac{11}{3} - \frac{8}{3}\right)^2 + \left(\frac{19}{3} - \frac{13}{3}\right)^2}$

Simplify each term:

$AB = \sqrt{\left(\frac{6}{3}\right)^2 + \left(\frac{3}{3}\right)^2 + \left(\frac{6}{3}\right)^2}$

$AB = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9}$

Thus:

$AB = 3$

Final Answer:

$Option (1) : \; 3$