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:

Proportional relationships often simplify to constants. Always isolate terms to identify constants or dependencies.

Answer 1

The Correct Option is D

The given condition states that \( (x + y) \) is proportional to \( (x - y) \), which means: \[ x + y = k (x - y), \] where \( k \) is a proportionality constant. Step 1: Simplify the equation. Rewriting the equation: \[ x + y = kx - ky. \] Rearranging terms: \[ x - kx = -ky - y. \] Factoring: \[ x(1 - k) = -y(1 + k). \] Step 2: Solve for \( \frac{x}{y} \). Divide both sides by \( y(1 - k) \) (assuming \( 1 - k \neq 0 \)): \[ \frac{x}{y} = -\frac{1 + k}{1 - k}. \] Step 3: Interpret the result. Since \( k \) is a constant, \( \frac{x}{y} \) is also a constant. It does not depend on \( x \) or \( y \) individually. Final Answer: \[ \boxed{\text{is a constant}} \]