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: Find the dividing point using the section formula.
With \(A(-3,-2,7)\) and \(B(3,1,-2)\), the point dividing AB in ratio 2:1 (from A towards B) is
\[P = \frac{2B+A}{3} = \frac{(6,2,-4)+(-3,-2,7)}{3} = \frac{(3,0,3)}{3} = (1,0,1)\]

Step 2: Get a simplified normal direction.
\(\vec{AB} = B-A = (6,3,-9)\), which simplifies (dividing by 3) to the direction \((2,1,-3)\), and this is the normal to the required plane since the plane is perpendicular to AB.

Step 3: Write the plane equation.
\[2(x-1) + 1(y-0) - 3(z-1) = 0 \implies 2x+y-3z+1=0\]

Step 4: Find the y-intercept.
Setting \(x=0, z=0\):
\[y+1=0 \implies y=-1\]
\[\boxed{-1}\]