Question

Equation of the plane passing through the point $(1, 2, 3)$ and parallel to the plane $2x + 3y - 4z = 0$ is

• A. $2x + 3y + 4z - 8 = 0$
• B. $2x + 3y - 4z + 4 = 0$
• C. $2x + 3y + 4z + 4 = 0$
• D. $2x + 3y + 4z = 20$

Hint:

Since parallel planes must share identical variables up to their constant terms, the correct answer must start with the exact sequence $2x + 3y - 4z$. Looking at the choices, only option (B) keeps these exact signs intact, allowing you to choose it instantly without doing any calculations!

Answer 1

The Correct Option is B

Step 1: Use the shared normal of parallel planes.
A plane parallel to $2x+3y-4z=0$ keeps the same normal $(2,3,-4)$, so it has the form $2x+3y-4z=d$ for some constant $d$.
Step 2: Impose the point condition.
The plane must contain $(1,2,3)$, so substitute these coordinates to find $d$.
Step 3: Compute $d$.
$d=2(1)+3(2)-4(3)=2+6-12=-4$.
Step 4: Write the plane.
$2x+3y-4z=-4$, i.e. $2x+3y-4z+4=0$.
Step 5: Verify with the point.
Plugging in $(1,2,3)$: $2+6-12+4=0$, confirming the point lies on it.
Step 6: Match the option.
This equals option (B). \[ \boxed{2x+3y-4z+4=0} \]