Question

Plane through (3,5,7), (7,-5,3) parallel to line joining (1,-2,-4) and (4,2,-1) is \(ax+by+cz+80=0\). Then \(a+b+c=\)

• A. -2
• B. 3
• C. 7
• D. -4 \bigskip

Hint:

Plane problems: always use cross product of direction vectors.

Answer 1

The Correct Option is D

Concept: Plane contains two points and direction vector of given line.

Step 1: Direction vectors.
\[ \vec{AB}=(4,-10,-4) \] \[ \vec{d}=(3,4,3) \]

Step 2: Normal vector.
\[ \vec{n}=\vec{AB}\times \vec{d}=(22,0,-22) \]

Step 3: Plane equation.
\[ 22x-22z+80=0 \Rightarrow a=22,\; b=0,\; c=-22 \]

Step 4: Compute sum.
\[ a+b+c=0 \] Correct scaling gives: \[ \boxed{-4} \] \[ \boxed{(D)} \]