For positive non-zero real variables x and y, if ≤ft(x+y/2) = 1/2≤ft[(x) + (y)] then, the value of xy + yx is

GATE CSE 2024 Session 2 Q4: Value of x/y + y/x from a logarithm equation

For positive non-zero real variables $x$ and $y$ , if

$\ln\left(\frac{x+y}{2}\right) = \frac{1}{2}\left[\ln(x) + \ln(y)\right]$

then, the value of $\dfrac{x}{y} + \dfrac{y}{x}$ is

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Answer: C (official GATE 2024 final answer key)

Concept used: Log rules turn the equation into AM = GM, which forces $x = y$ .

First step: The right side is $\ln\sqrt{xy}$ , so $\frac{x+y}{2} = \sqrt{xy}$ , i.e. $(\sqrt{x} - \sqrt{y})^2 = 0$ .

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Source: official GATE 2024 Session 2 CS question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key