Question

Let \(A = (-3, -2, 7)\) and \(B = (3, 1, -2)\). Let a plane perpendicular to the line segment AB divide AB in the ratio 2:1. Then the intercept made by the plane on y-axis is

• A. \(-\frac{1}{2}\)
• B. \(\frac{1}{3}\)
• C. 2
• D. -1

Hint:

For plane perpendicular to line segment, use point on plane and dot product with direction vector to find equation; then find intercepts by setting other coordinates 0.

Answer 1

The Correct Option is D

Step 1: Determine point dividing AB in 2:1.
Using section formula, point \(P\) dividing AB in ratio 2:1: \[ P = \left( \frac{2 \cdot 3 + 1 \cdot (-3)}{2+1}, \frac{2 \cdot 1 + 1 \cdot (-2)}{2+1}, \frac{2 \cdot (-2) + 1 \cdot 7}{2+1} \right) = (1, 0, 1) \]

Step 2: Direction vector of AB.
\(\vec{AB} = B - A = (6, 3, -9)\)

Step 3: Plane perpendicular to AB.
Equation: \(6(x - 1) + 3(y - 0) - 9(z - 1) = 0 \implies 2(x - 1) + (y - 0) - 3(z - 1) = 0\)

Step 4: Find y-intercept.
Set \(x = 0, z = 0\): \(2(-1) + y - 3(-1) = 0 \implies -2 + y + 3 = 0 \implies y = -1\)

Step 5: Final conclusion.
Hence, y-intercept = \[ \boxed{-1} \]