Find the equation of the plane that passes through the point $(1, 2, 3)$ and is parallel to the plane $2 x - 3 y + z = 5$ .

easy 1 mark Class 12 · Mathematics · Three Dimensional Geometry

Answer

2x - 3y + z = 1

The correct answer is 2x - 3y + z = 1 — Thus the required plane is $2 x - 3 y + z = 1$ .

Step-by-step solution

Identify the normal vector of the given plane, which is $(2, - 3, 1)$ .
Use the point–normal form of a plane: $2 (x - 1) - 3 (y - 2) + 1 (z - 3) = 0$ .
Expand and simplify: $2 x - 2 - 3 y + 6 + z - 3 = 0$ gives $2 x - 3 y + z = 1$ .
Thus the required plane is $2 x - 3 y + z = 1$ .

Common mistake: Using a different normal vector or forgetting the point‑normal form

Concept tested: Plane equation from point and normal vector