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:
We are given a 3D straight line in symmetrical form and told that it lies entirely within the flat surface of the plane $x + 3y - \alpha z + \beta = 0$. We need to solve for the product of the structural parameters $\alpha$ and $\beta$.

Step 2: Key Formula or Approach:
For a straight line passing through point $P(x_1, y_1, z_1)$ with direction ratios $(l, m, n)$ to lie completely inside a plane $Ax + By + Cz + D = 0$, two independent constraints must be met simultaneously: 1. The direction vector of the line must be perpendicular to the normal vector of the plane: $Al + Bm + Cn = 0$. 2. Any point on the line, particularly the reference point $P(x_1, y_1, z_1)$, must satisfy the plane equation: $Ax_1 + By_1 + Cz_1 + D = 0$.

Step 3: Detailed Explanation:
Extract the properties of the line from its given equation: $$\text{Passing point: } P(2, 1, -2)$$ $$\text{Direction ratios: } (l, m, n) = (3, -5, 2)$$ The coefficients of our plane equation give its normal vector coordinates: $(A, B, C) = (1, 3, -\alpha)$. Apply the first constraint (perpendicular vectors): $$A\cdot l + B\cdot m + C\cdot n = 0$$ $$(1)(3) + (3)(-5) + (-\alpha)(2) = 0$$ $$3 - 15 - 2\alpha = 0$$ $$-12 - 2\alpha = 0 \implies 2\alpha = -12 \implies \alpha = -6$$ Now substitute $\alpha = -6$ back into the plane formula to update it: $$x + 3y - (-6)z + \beta = 0 \implies x + 3y + 6z + \beta = 0$$ Apply the second constraint by plugging the coordinates of point $P(2, 1, -2)$ into this updated plane equation: $$(2) + 3(1) + 6(-2) + \beta = 0$$ $$2 + 3 - 12 + \beta = 0$$ $$-7 + \beta = 0 \implies \beta = 7$$ We need to calculate the product value $\alpha \beta$: $$\alpha \beta = (-6) \times (7) = -42$$

Step 4: Final Answer:
The product value $\alpha \beta$ is equal to $-42$, which corresponds to option (C).