Key Concepts and Formulas

To solve this problem, we will utilize two fundamental concepts from 3D Geometry:

1. Equation of a Plane Parallel to a Given Plane: If the equation of a plane is given by $Ax+By+Cz+D=0$, any plane parallel to it will share the same normal vector. Consequently, its equation can be expressed in the form $Ax+By+Cz+d=0$, where $d$ is a constant that determines the specific position of the parallel plane. The coefficients $A, B, C$ define the direction of the normal vector.
2. Distance from a Point to a Plane: The perpendicular distance $P$ from a point $(x_0, y_0, z_0)$ to a plane $Ax+By+Cz+D=0$ is given by the formula: $P = \frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}$ The absolute value in the numerator ensures that the distance is always non-negative.

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Step-by-Step Solution

Step 1: Identify the Normal Vector of the Given Plane

The equation of the given plane is: $x - 2y + 2z - 5 = 0$ From the standard form $Ax+By+Cz+D=0$, we can identify the coefficients: $A=1$, $B=-2$, and $C=2$. These coefficients represent the components of the normal vector to the plane, $\vec{n} = \langle 1, -2, 2 \rangle$. This normal vector is crucial because all planes parallel to the given plane will have the same normal vector direction.

Step 2: Formulate the General Equation of a Parallel Plane

Since the required plane is parallel to the given plane, it must have the same normal vector. This means the coefficients of $x, y,$ and $z$ in its equation will be identical to those of the given plane. Only the constant term will differ. Therefore, the general equation of any plane parallel to $x - 2y + 2z - 5 = 0$ can be written as: $x - 2y + 2z + d = 0$ where $d$ is an unknown constant that we need to determine.

Step 3: Apply the Distance Condition from the Origin

We are given that the required plane is at a unit distance (i.e., $P=1$) from the origin. The origin is the point $(x_0, y_0, z_0) = (0,0,0)$. Now, we use the distance formula from the point $(0,0,0)$ to the plane $x - 2y + 2z + d = 0$. Here, $A=1$, $B=-2$, $C=2$, and the constant term is $d$. Substituting these values into the distance formula: $P = \frac{|A(x_0)+B(y_0)+C(z_0)+d|}{\sqrt{A^2+B^2+C^2}}$ $1 = \frac{|1(0) - 2(0) + 2(0) + d|}{\sqrt{1^2 + (-2)^2 + 2^2}}$

Step 4: Solve for the Constant 'd'

Let's simplify the equation obtained in Step 3: $1 = \frac{|0 - 0 + 0 + d|}{\sqrt{1 + 4 + 4}}$ $1 = \frac{|d|}{\sqrt{9}}$ $1 = \frac{|d|}{3}$ To find the value(s) of $d$, we multiply both sides by 3: $|d| = 3$ This equation implies that $d$ can be either $3$ or $-3$. This is geometrically significant: it means there are two distinct planes parallel to the given plane that are at a unit distance from the origin – one on each side of the origin.

Step 5: Write the Equations of the Possible Planes

We substitute the two possible values of $d$ back into the general equation of the parallel plane ($x - 2y + 2z + d = 0$):

• For $d = 3$: $x - 2y + 2z + 3 = 0$
• For $d = -3$: $x - 2y + 2z - 3 = 0$

Step 6: Compare with the Given Options

Now, we compare these two derived plane equations with the provided options: (A) $x-2y+2z-3=0$ (B) $x-2y+2z+1=0$ (C) $x-2y+2z-1=0$ (D) $x-2y+2z+5=0$

We observe that the equation $x - 2y + 2z - 3 = 0$ perfectly matches option (A).

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Common Mistakes & Tips

• Absolute Value: Always include the absolute value in the numerator of the distance formula ($|Ax_0+By_0+Cz_0+D|$). Forgetting it might lead to only one solution for the constant $d$, when often two solutions (representing two planes) exist.
• Magnitude Calculation: Be careful when calculating the denominator $\sqrt{A^2+B^2+C^2}$. Arithmetic errors here are common and will propagate through the solution.
• Understanding "Parallel": Remember that parallel planes have the same normal vector direction, meaning their $x, y, z$ coefficients are proportional (or identical, if the equation is normalized). Only the constant term changes.
• "Unit Distance from Origin": This specifically means the point is $(0,0,0)$ and the distance $P=1$. Do not confuse it with distance from other points or different distance values.

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Summary

This problem effectively tests your understanding of the properties of parallel planes and the formula for the distance from a point to a plane. The solution involves first establishing the general form of a plane parallel to the given one by retaining its normal vector. Then, the distance formula is applied, using the origin as the reference point and the given unit distance, to solve for the unknown constant. The absolute value in the distance formula is crucial as it correctly yields two possible planes, one of which matches the given options.

The final answer is $\boxed{\text{A}}$.