3D Geometry
Plane through perpendicular intersection; direction cosines
nta_pyq_2023_jan
Grade 12
Question:
If the equation of the plane passing through the point $(1,1,2)$ and perpendicular to the line $x - 3y + 2z - 1 = 0$, $4x - y + z = 0$ is $Ax + By + Cz = 1$, then $140(C - B + A)$ is equal to _____.
Step-by-Step Solution
Key Concept: Find the direction of the line (cross product of normals of the two planes), use it as normal to desired plane, write plane through given point.
Plane: $-x+7y+11z=28$. $A=-1/28, B=7/28, C=11/28$. $140(C-B+A) = 140\left(\frac{11-7-1}{28}\right) = 140 \cdot \frac{3}{28} = 15$. Answer: 15
Correct Answer: 15