If (x + y) Is Proportional to (x − y), Is x/y Constant?

This problem is a classic GATE aptitude and competitive exam question.
We analyze whether the ratio x/y remains constant when
(x + y) is proportional to (x − y).

Problem Statement

Given two distinct, non-zero real variables x and y,
such that:

(x + y) ∝ (x − y)

Determine whether the ratio x/y is constant.

Understanding Proportionality

Proportionality means there exists a non-zero constant k such that:

x + y = k(x − y)

Mathematical Proof

Starting from the proportionality equation:

x + y = k(x − y)

Expanding the right-hand side:

x + y = kx − ky

Rearranging terms:

x − kx = −ky − y

Taking common factors:

x(1 − k) = −y(k + 1)

Dividing both sides by y(1 − k):

x / y = −(k + 1) / (1 − k) = (k + 1) / (k − 1)

Conclusion from the Proof

The ratio x/y depends only on the proportionality constant k,
not on the individual values of x or y.

Since k is constant, the ratio x/y is also constant.
The condition that x and y are distinct and non-zero
ensures k ≠ 1, avoiding division by zero.

Correct Answer

(D) x/y is a constant

The ratio is given by:

x / y = (k + 1) / (k − 1)

Option-wise Analysis

(A) Depends on xy

❌ Incorrect. The product xy does not appear in the final expression.
The ratio depends only on k.

(B) Depends only on x and not on y

❌ Wrong. The derivation eliminates both x and y;
neither appears individually in the final result.

(C) Depends only on y and not on x

❌ Incorrect for the same reason as option (B).

(D) Is a constant

✅ Correct. The ratio x/y is fully determined by the constant k.

Final Answer

The correct option is (D): x/y is a constant.