Question

Equation of planes parallel to the plane $x - 2y + 2z + 4 = 0$ which are at a distance of one unit from the point $(1, 2, 3)$ are

• A. $x + 2y + 2z = 6$, $x + 2y + 2z = 0$
• B. $x - 2y + 2z = 0$, $x - 2y + 2z - 6 = 0$
• C. $x - 2y - 6 = 0$, $x - 2y + z = 6$
• D. $x + 2y + 2z = -6$, $x + 2y + 2z = 5$

Hint:

Parallel planes must have identical variable coefficients ($x - 2y + 2z$). Looking at the choices, options (A) and (D) change the sign of $y$ to $+2y$, and option (C) alters the $z$ term completely. This leaves option (B) as the only option that keeps the correct parallel direction coefficients!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find the mathematical equations of two planes that are parallel to the given reference plane $x - 2y + 2z + 4 = 0$, such that their perpendicular distance to the point $(1, 2, 3)$ is exactly 1 unit.

Step 2: Key Formula or Approach:
1. Any plane parallel to $Ax + By + Cz + D = 0$ shares identical direction normal coefficients and can be written as: $$Ax + By + Cz + \lambda = 0$$ 2. The formula for the perpendicular distance $d$ from a point $(x_1, y_1, z_1)$ to a plane is: $$d = \frac{|Ax_1 + By_1 + Cz_1 + \lambda|}{\sqrt{A^2 + B^2 + C^2}}$$

Step 3: Detailed Explanation:
3. Write down the expression for a parallel plane template: $$x - 2y + 2z + \lambda = 0$$ 4. Substitute the given point $(1, 2, 3)$ and set distance $d = 1$: $$1 = \frac{|1(1) - 2(2) + 2(3) + \lambda|}{\sqrt{1^2 + (-2)^2 + 2^2}}$$ $$1 = \frac{|1 - 4 + 6 + \lambda|}{\sqrt{1 + 4 + 4}}$$ $$1 = \frac{|3 + \lambda|}{\sqrt{9}} = \frac{|3 + \lambda|}{3}$$ 5. Clear the denominator by multiplying across: $$|3 + \lambda| = 3$$ 6. Solve the absolute value equation, which splits into two cases: Case 1: $3 + \lambda = 3 \implies \lambda = 0$ Case 2: $3 + \lambda = -3 \implies \lambda = -6$ 5. Plug these values back into our parallel plane template: For $\lambda = 0$: $x - 2y + 2z = 0$ For $\lambda = -6$: $x - 2y + 2z - 6 = 0$

Step 4: Final Answer:
The equations of the parallel planes correspond to option (B).