Question:
Equation of the plane passing through the point $(1, 2, 3)$ and parallel to the plane $2x + 3y - 4z = 0$ is
Equation of the plane passing through the point $(1, 2, 3)$ and parallel to the plane $2x + 3y - 4z = 0$ is
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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!
Updated On: Jun 11, 2026
- $2x + 3y + 4z - 8 = 0$
- $2x + 3y - 4z + 4 = 0$
- $2x + 3y + 4z + 4 = 0$
- $2x + 3y + 4z = 20$
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Question:
The problem requires finding the coordinate equation of a 3D plane that passes through a specific point $(x_1, y_1, z_1) = (1, 2, 3)$ and is oriented perfectly parallel to an existing reference plane $2x + 3y - 4z = 0$.
Step 2: Key Formula or Approach:
Parallel planes share identical direction normal vectors. Therefore, any plane parallel to the plane $Ax + By + Cz + D = 0$ can be written in the form: $$Ax + By + Cz + k = 0$$ Where $k$ is a scalar constant determined by substituting the given point coordinates.
Step 3: Detailed Explanation:
The given plane is $2x + 3y - 4z = 0$. Any plane parallel to it shares the same leading coefficients: $$2x + 3y - 4z + k = 0$$ Since this target plane passes through the specific point $(1, 2, 3)$, these coordinates must satisfy the equation. Substitute $x = 1, y = 2,$ and $z = 3$: $$2(1) + 3(2) - 4(3) + k = 0$$ $$2 + 6 - 12 + k = 0$$ $$-4 + k = 0 \implies k = 4$$ Substitute $k = 4$ back into our parallel template equation: $$2x + 3y - 4z + 4 = 0$$ This matches the formula given in option (B).
Step 4: Final Answer:
The equation of the parallel plane is $2x + 3y - 4z + 4 = 0$, which corresponds to option (B).
The problem requires finding the coordinate equation of a 3D plane that passes through a specific point $(x_1, y_1, z_1) = (1, 2, 3)$ and is oriented perfectly parallel to an existing reference plane $2x + 3y - 4z = 0$.
Step 2: Key Formula or Approach:
Parallel planes share identical direction normal vectors. Therefore, any plane parallel to the plane $Ax + By + Cz + D = 0$ can be written in the form: $$Ax + By + Cz + k = 0$$ Where $k$ is a scalar constant determined by substituting the given point coordinates.
Step 3: Detailed Explanation:
The given plane is $2x + 3y - 4z = 0$. Any plane parallel to it shares the same leading coefficients: $$2x + 3y - 4z + k = 0$$ Since this target plane passes through the specific point $(1, 2, 3)$, these coordinates must satisfy the equation. Substitute $x = 1, y = 2,$ and $z = 3$: $$2(1) + 3(2) - 4(3) + k = 0$$ $$2 + 6 - 12 + k = 0$$ $$-4 + k = 0 \implies k = 4$$ Substitute $k = 4$ back into our parallel template equation: $$2x + 3y - 4z + 4 = 0$$ This matches the formula given in option (B).
Step 4: Final Answer:
The equation of the parallel plane is $2x + 3y - 4z + 4 = 0$, which corresponds to option (B).
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