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 $\dfrac{x}{y}$ 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:

Whenever "proportional" appears, introduce a constant $k$. Solving usually reveals whether a ratio is constant or variable.

Answer 1

The Correct Option is D

Step 1: Proportional relation.
We are given: \[ (x+y) \propto (x-y) \] This means there exists a constant $k$ such that \[ x + y = k(x - y) \] 

Step 2: Simplify the equation.
\[ x + y = kx - ky \] Rearranging: \[ x - kx = -ky - y \] \[ x(1-k) = -y(k+1) \] 

Step 3: Solve for \(\dfrac{x}{y}\)

\[ \frac{x}{y} = \frac{-(k+1)}{(1-k)} \] 

Step 4: Interpretation.
The ratio $\dfrac{x}{y}$ is expressed purely in terms of the proportionality constant $k$. Since $k$ is fixed (independent of $x$ and $y$ individually), the value of $\dfrac{x}{y}$ is a constant. \[ \boxed{\text{constant}} \]