Question

The line $\frac{x-2}{3} = \frac{y-1}{-5} = \frac{z+2}{2}$ lies in the plane $x + 3y - \alpha z + \beta = 0$, then value of $\alpha \beta$ is

• A. 42
• B. 1
• C. $-42$
• D. $-2$

Hint:

Always apply the direction dot-product condition ($\vec{d} \cdot \vec{n} = 0$) first. It is an isolated equation containing only one parameter ($\alpha$), letting you solve for it directly before involving the translation constant $\beta$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
A line lies completely in a plane; find the product αβ given the line and plane equations.

Step 2: Key Formula or Approach:
For a line to lie in a plane: (1) direction vector ⟂ normal vector, (2) a point on the line satisfies the plane.

Step 3: Detailed Explanation:
Point P(2,1,–2), direction (3,–5,2), normal (1,3,–α). Condition 1: 3–15–2α=0 → α=–6. Condition 2: 2+3+6(–2)+β=0 → –7+β=0 → β=7. Product αβ = (–6)(7) = –42.

Step 4: Final Answer:
αβ = –42, matching option (C).