GATE CS 2024 Set 2 — Question 4

MCQ+1 / -0.33EasyRatios, Percentages, Powers, Exponents & LogarithmsQuantitative AptitudeGeneral Aptitude

General Aptitude → Quantitative Aptitude → Ratios, Percentages, Powers, Exponents & Logarithms

Last updated 20 August 2026

Question

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}

1/2

Given equation:

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

Using logarithmic properties

\ln(a) + \ln(b) = \ln(ab)

and

k \ln(a) = \ln(a^k)

\ln \left(\frac{x+y}{2}\right) = \frac{1}{2} \ln (xy) = \ln ((xy)^{1/2}) = \ln (\sqrt{xy})

Taking antilog on both sides:

\frac{x+y}{2} = \sqrt{xy}

x+y = 2\sqrt{xy}

Squaring both sides:

(x+y)^2 = (2\sqrt{xy})^2

x^2 + y^2 + 2xy = 4xy

x^2 + y^2 - 2xy = 0

(x-y)^2 = 0

x = y

We need to find the value of

\frac{x}{y} + \frac{y}{x}

. Since

x=y

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