GATE CS 2024 Set 2 Q.4: Quantitative Aptitude

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

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

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

$1/2$

Official answer and solution

Solution

Use logarithm properties to equate $\frac{x+y}{2}$ and $\sqrt{xy}$ , showing $x = y$ .

Using the logarithm identities $\ln(x) + \ln(y) = \ln(xy)$ and $\frac{1}{2}\ln(xy) = \ln(\sqrt{xy})$ :

\ln\left(\frac{x+y}{2}\right) = \ln(\sqrt{xy}) \implies \frac{x+y}{2} = \sqrt{xy}

Squaring both sides and rearranging:

(x+y)^2 = 4xy \implies x^2 + 2xy + y^2 - 4xy = 0 \implies (x - y)^2 = 0 \implies x = y

Therefore:

\frac{x}{y} + \frac{y}{x} = 1 + 1 = 2

Final answer

Option (C)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.