Question

A line passes through (2, 1, 3) and (1, 2, -1), then
(A) Equation is \( \frac{x-2}{1} = \frac{y-1}{1} = \frac{z-3}{4} \)
(B) Equation is \( \frac{x+2}{-1} = \frac{y+1}{1} = \frac{z+3}{4} \)
(C) Equation is \( \vec{r} = 2\vec{i} + \vec{j} + 3\vec{k} + \lambda(\vec{i} - \vec{j} + 4\vec{k}) \)
(D) Equation is \( \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z+1}{4} \)
Choose the correct answer:

• A. (A), (B) and (D) only
• B. (B) and (C) only
• C. (A), (C) and (D) only
• D. (C) and (D) only

Hint:

Different points on the same line can yield different-looking equations, so always verify by plugging in the points.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A line through \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) has direction ratios: \( (x_1-x_2, y_1-y_2, z_1-z_2) \).

Step 2: Detailed Explanation:
Direction ratios \( = (2-1, 1-2, 3-(-1)) = (1, -1, 4) \).
Cartesian equation using \( (1, 2, -1) \): \( \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z+1}{4} \) (Matches D).
Vector equation using \( (2, 1, 3) \): \( \vec{r} = (2\vec{i} + \vec{j} + 3\vec{k}) + \lambda(\vec{i} - \vec{j} + 4\vec{k}) \) (Matches C).

Step 3: Final Answer:
Statements (C) and (D) are correct.