Write the equation of the plane that passes through the point $(2,, - 1,, 4)$ and is parallel to the plane $4 x - y + 2 z = 7$ .

medium 3 marks Class 12 · Mathematics · Three Dimensional Geometry

Answer

4 x - y + 2 z = 17

The correct answer is $4 x - y + 2 z = 17$ — Expand to obtain $4 x - y + 2 z - 17 = 0$ or equivalently $4 x - y + 2 z = 17$ .

Step-by-step solution

The normal vector of the given plane is $n = (4,, - 1,, 2)$ .
Since the required plane is parallel, it has the same normal vector.
Use the point‑normal form: $4 (x - 2) - 1 (y + 1) + 2 (z - 4) = 0$ .
Expand to obtain $4 x - y + 2 z - 17 = 0$ or equivalently $4 x - y + 2 z = 17$ .

Common mistake: Incorrect sign while expanding the point–normal form

Concept tested: Plane through point parallel to given plane