If two distinct non-zero real variables x and y are such that (x + y) is proportional
to (x - y) then the value of
x
y
Options:
(A) depends on xy
(B) depends only on x and not on y
(C) depends only on y and not on x
(D) is a constant
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2024 Q.2)**
**1. Governing Concept**:
- **Topic**: Quantitative Aptitude & Direct Proportionality
- If $(x + y) \propto (x - y)$, then:
$x + y = k(x - y)$
where $k$ is a non-zero proportionality constant ($k \neq 1$ since $y \neq 0$).
$x + y = kx - ky \implies y(1 + k) = x(k - 1) \implies \frac{x}{y} = \frac{k + 1}{k - 1}$
- Since $k$ is a constant, the ratio $\frac{x}{y}$ is also an invariant constant.
**2. Option Analysis**:
- **Option (A) [depends on xy] is INCORRECT**: The ratio is strictly independent of the product $xy$.
- **Option (B) [depends only on x and not on y] is INCORRECT**: The individual variables cancel out completely in the ratio.
- **Option (C) [depends only on y and not on x] is INCORRECT**: Neither individual variable determines the ratio.
- **Option (D) [is a constant] is CORRECT**: The ratio $\frac{x}{y} = \frac{k+1}{k-1}$ is purely a constant.
**3. Final Answer**:
- **Correct Choice**: **D**
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