Question

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:

• 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

Hint:

To solve proportionality problems:
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.

Answer 1

The Correct Option is D

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 \( \mathbf{(D)} \).