A equation of a plane parallel to the plane x - 2y + 2z - 5 = 0 and at a unit distance from the orig

Q: Wait, why do we assume the plane is $x - 2y + 2z + d = 0$ instead of $k(x - 2y + 2z) - 5 = 0$?

That is a fantastic question! Actually, both approaches are mathematically sound. By writing the plane as $x - 2y + 2z + d = 0$, we are essentially saying that the plane is a translation of the original plane. The constant $d$ captures all possible parallel planes. If you used $k(x - 2y + 2z) - 5 = 0$, you would eventually arrive at the same result, but the $d$ method is generally cleaner and less prone to algebraic errors.

Q: Why is there an absolute value in the distance formula?

The absolute value is crucial because distance is a physical quantity—it must always be positive. When we calculate the distance from the origin to a plane, the expression $ax + by + cz + d$ can be positive or negative depending on which side of the origin the plane lies. The absolute value ensures that we are measuring the magnitude of the displacement, not the direction.

Q: Why are there two possible answers, but only one is in the options?

You have hit on a classic JEE trap! Mathematically, there are indeed two planes parallel to the given plane at a unit distance from the origin: one on the positive side and one on the negative side. When you solve $|d| = 3$, you get $d = 3$ and $d = -3$. This leads to $x - 2y + 2z + 3 = 0$ and $x - 2y + 2z - 3 = 0$. Since only one of these appears in your multiple-choice options, you simply select the one that matches. Always look for both possibilities!

Question

A equation of a plane parallel to the plane $x - 2y + 2z - 5 = 0$ and at a unit distance from the origin is :

Options:

(A) $x - 2y + 2z - 3 = 0$

(B) $x - 2y + 2z + 1 = 0$

(C) $x - 2y + 2z - 1 = 0$

(D) $x - 2y + 2z + 5 = 0$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a vast, empty 3D coordinate system. You have a plane, a flat, infinite sheet of paper, defined by the equation $x - 2y + 2z - 5 = 0$.

This plane has a specific orientation, a tilt, defined by its normal vector. By looking at the coefficients of $x$, $y$, and $z$, we can immediately identify this vector as $\vec{n} = \hat{i} - 2\hat{j} + 2\hat{k}$.

### The Concept of Parallelism

We need to find a new plane that is parallel to our original one. For two planes to be parallel, they must share the exact same normal vector $\vec{n}$.

Because their orientation is identical, the only thing that can change is their position in space. In the equation of a plane, this position is dictated solely by the constant term.

We can confidently write the equation of our new, unknown plane as $x - 2y + 2z + d = 0$, where $d$ is the mystery constant we need to uncover.

### Bridging Algebra and Geometry

The problem provides us with a geometric constraint: this new plane is exactly one unit away from the origin, $O(0,0,0)$.

We utilize the perpendicular distance formula. The distance $P$ from the origin to a plane $ax + by + cz + d = 0$ is given by:

$$P = \frac{|d|}{\sqrt{a^2 + b^2 + c^2}}$$

We know $P = 1$, and our coefficients are $a=1$, $b=-2$, and $c=2$. Substituting these into our formula, we get:

$$\frac{|d|}{\sqrt{1^2 + (-2)^2 + 2^2}} = 1$$

### The Final Calculation

The denominator is $\sqrt{1 + 4 + 4}$, which simplifies to $\sqrt{9}$, or $3$. Our equation becomes:

$$\frac{|d|}{3} = 1$$

Multiplying both sides by $3$, we find that $|d| = 3$. The absolute value indicates that $d$ can be either $3$ or $-3$.

This makes perfect sense geometrically, as there is one plane at a distance of one unit on one side of the origin, and another plane at a distance of one unit on the opposite side.

Substituting these back into our general equation, we get two possible planes: $x - 2y + 2z + 3 = 0$ and $x - 2y + 2z - 3 = 0$.

The correct choice is **$x - 2y + 2z - 3 = 0$**.

Animated Solution for Mathematics - Three Dimensional Geometry: A equation of a plane parallel to the plane $x - 2 y + 2 z - 5 = 0$ and at a unit distance from the origin is :

Select Answer:

Visualized Solution

The Given Plane

Parallel Planes Concept

Equation of Parallel Plane

Distance from Origin

Distance Formula

Substituting Values

Evaluating the Denominator

Simplifying the Root

Solving for $d$

Final Equations

The Sigma Insight: Equation of a Plane

Analyzing the Setup

The Concept of Parallelism

Bridging Algebra and Geometry

The Final Calculation

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Animated Solution for Mathematics - Three Dimensional Geometry: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A equation of a plane parallel to the plane x−2y+2z−5=0 and at a unit distance from the origin is :