If two distinct non-zero real variables \( x \) and \( y \) are such that \( (x + y) \) is proportional to \( (x - y) \), then the value of \( \frac{x}{y} \) is:
Show Hint
1. Replace proportionality with an equation involving a constant.
2. Simplify and rearrange terms to find the relationship between variables.
3. Evaluate the dependence of the result on given terms or constants.
- depends on \( xy \)
- depends only on \( x \) and not on \( y \)
- depends only on \( y \) and not on \( x \)
- is a constant
The Correct Option is D
Solution and Explanation
Step 1: Write the proportionality relation. The question states that \( (x + y) \propto (x - y) \). This implies that: \[ x + y = k(x - y) \] where \( k \) is a constant of proportionality.
Step 2: Simplify the equation. Rewriting the equation: \[ x + y = kx - ky \] Rearranging terms: \[ x - kx = -ky - y \] \[ x(1 - k) = -y(1 + k) \] \[ \frac{x}{y} = \frac{- (1 + k)}{1 - k} \]
Step 3: Analyze the result. The value of \( \frac{x}{y} \) depends only on \( k \), which is a constant. Thus, \( \frac{x}{y} \) is also a constant.
Conclusion: The value of \( \frac{x}{y} \) is \( \mathbf{constant} \), corresponding to option \( \mathbf{(D)} \).
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