Discussion
If two distinct non-zero real variables \(x\) and \(y\) are such that \((x + y)\) is proportional to \((x - y)\) then the value of \(\frac{x}{y}\)
Answer: D
The ratio \(\frac{x}{y}\) is a constant.
Proportionality means \(x + y = k(x - y)\) for a fixed constant \(k\). Collect the terms:
\[ \begin{aligned} x + y &= kx - ky \\ y(1 + k) &= x(k - 1) \\ \frac{x}{y} &= \frac{k + 1}{k - 1} \end{aligned} \]
The right-hand side contains only \(k\), so \(x/y\) has the same value whatever \(x\) and \(y\) are, provided they satisfy the proportionality. (\(k = 1\) would force \(y = 0\), which the question excludes.) It does not depend on \(xy\), on \(x\) alone or on \(y\) alone.
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