GATE CS 2024 Set 1 | Question 2

MCQ · 1 mark · −0.33

If two distinct non-zero real variables \(x\) and \(y\) are such that \((x + y)\) is proportional to \((x - y)\) then the value of \(\frac{x}{y}\)

  1. Option A: depends on \(xy\)
  2. Option B: depends only on \(x\) and not on \(y\)
  3. Option C: depends only on \(y\) and not on \(x\)
  4. Option D: is a constant (correct)

Answer

(D)

Explanation

“Proportional” means \(x + y = k(x - y)\) for some fixed constant \(k\). Rearranging,

\[x + y = kx - ky \;\Rightarrow\; y(k + 1) = x(k - 1) \;\Rightarrow\; \frac{x}{y} = \frac{k + 1}{k - 1}.\]

The right-hand side involves only the constant \(k\), so \(\frac{x}{y}\) is the same for every pair that satisfies the condition: it is a constant. (Here \(k \neq \pm 1\), because \(k = 1\) would force \(y = 0\) and \(k = -1\) would force \(x = 0\).)

Answer: D