Key Concepts and Formulas

• Equation of a Plane in Intercept Form: If a plane intersects the x-axis, y-axis, and z-axis at distances $a$, $b$, and $c$ respectively from the origin, its equation can be expressed as: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
• Perpendicular Distance of a Plane from the Origin: For a plane given by the general equation $Ax + By + Cz + D = 0$, the perpendicular distance $p$ from the origin $(0,0,0)$ to the plane is calculated using the formula: $p = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}}$
• Invariance of Perpendicular Distance: The perpendicular distance of a plane from a fixed point (like the origin) is an intrinsic geometric property. It remains constant regardless of the orientation of the coordinate axes, provided the origin itself does not change.

Step-by-Step Solution

The problem describes a single plane that is viewed from two different rectangular coordinate systems, both sharing the same origin. Our strategy will be to calculate the perpendicular distance of this plane from the origin in both coordinate systems and then equate these distances.

Step 1: Formulate the Plane Equation in the First Coordinate System

• What we are doing: We are writing the equation of the plane based on its intercepts in the first coordinate system.
• Why we are doing it: The problem provides the intercepts directly ($a, b, c$), making the intercept form the most straightforward way to represent the plane.

Given that the plane cuts the axes at distances $a, b, c$ from the origin in the first system, its equation is: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ To use the perpendicular distance formula, we convert this into the general form $Ax + By + Cz + D = 0$: $\left(\frac{1}{a}\right)x + \left(\frac{1}{b}\right)y + \left(\frac{1}{c}\right)z - 1 = 0$ Here, we identify $A = \frac{1}{a}$, $B = \frac{1}{b}$, $C = \frac{1}{c}$, and $D = -1$.

Step 2: Calculate Perpendicular Distance from Origin (First System)

• What we are doing: We are calculating the shortest distance from the origin to the plane defined in Step 1.
• Why we are doing it: This perpendicular distance is a unique characteristic of the plane relative to the origin, which will be invariant across different coordinate systems with the same origin.

Using the distance formula $p = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}}$: $p = \frac{|-1|}{\sqrt{\left(\frac{1}{a}\right)^2 + \left(\frac{1}{b}\right)^2 + \left(\frac{1}{c}\right)^2}}$ $p = \frac{1}{\sqrt{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}}}$

Step 3: Formulate the Plane Equation in the Second Coordinate System

• What we are doing: We are writing the equation of the same plane using the intercepts given for the second coordinate system.
• Why we are doing it: We need to express the plane's properties in the second system to eventually compare them with the first system.

The problem states that the same plane cuts the axes at distances $a', b', c'$ in the second coordinate system. Its equation is: $\frac{x}{a'} + \frac{y}{b'} + \frac{z}{c'} = 1$ Converting to the general form: $\left(\frac{1}{a'}\right)x + \left(\frac{1}{b'}\right)y + \left(\frac{1}{c'}\right)z - 1 = 0$ Here, $A' = \frac{1}{a'}$, $B' = \frac{1}{b'}$, $C' = \frac{1}{c'}$, and $D' = -1$.

Step 4: Calculate Perpendicular Distance from Origin (Second System)

• What we are doing: We are calculating the perpendicular distance from the origin to the plane as described in the second coordinate system.
• Why we are doing it: This gives us another expression for the same intrinsic distance, which we will use for comparison.

Using the distance formula $p' = \frac{|D'|}{\sqrt{A'^2 + B'^2 + C'^2}}$: $p' = \frac{|-1|}{\sqrt{\left(\frac{1}{a'}\right)^2 + \left(\frac{1}{b'}\right)^2 + \left(\frac{1}{c'}\right)^2}}$ $p' = \frac{1}{\sqrt{\frac{1}{a'^2} + \frac{1}{b'^2} + \frac{1}{c'^2}}}$

Step 5: Equate the Distances and Simplify to Find the Relation

• What we are doing: We are equating the two expressions for the perpendicular distance obtained in Step 2 and Step 4.
• Why we are doing it: Since it is the same plane and both coordinate systems share the same origin, the perpendicular distance from the origin to the plane must be identical, regardless of which axis system is used. This is the core principle that links the two descriptions.

Therefore, $p = p'$: $\frac{1}{\sqrt{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}}} = \frac{1}{\sqrt{\frac{1}{a'^2} + \frac{1}{b'^2} + \frac{1}{c'^2}}}$ To simplify, we can square both sides: $\frac{1}{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}} = \frac{1}{\frac{1}{a'^2} + \frac{1}{b'^2} + \frac{1}{c'^2}}$ Now, take the reciprocal of both sides: $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} = \frac{1}{a'^2} + \frac{1}{b'^2} + \frac{1}{c'^2}$ Rearranging the terms to match the given options: $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} - \frac{1}{a'^2} - \frac{1}{b'^2} - \frac{1}{c'^2} = 0$

Common Mistakes & Tips

• Confusing Intercepts with Perpendicular Distance: Remember that $a, b, c$ are the intercepts (distances along the axes), not the perpendicular distance of the plane from the origin. The perpendicular distance is calculated using a specific formula involving these intercepts.
• Ignoring "Same Origin": The condition that both coordinate systems share the same origin is crucial. If the origins were different, the perpendicular distances from each origin to the plane would generally not be equal.
• Understanding Invariance: The key insight is that the physical distance from a point (the origin) to a plane is a geometric invariant. It does not change if you simply rotate your coordinate axes around that point. This makes equating the distances derived from different systems valid.

Summary

This problem leverages the concept that the perpendicular distance of a plane from a fixed origin is an invariant property, independent of the orientation of the coordinate axes. By first expressing the plane's equation in intercept form for two different coordinate systems (both sharing the same origin) and then converting them to the general form, we can apply the formula for the perpendicular distance from the origin. Equating these two distance expressions, which represent the same physical distance, leads directly to the required relationship between the intercepts.

The final answer is \boxed{A}