Question

If $\ln \left(\frac{a+b}{3}\right)=\left(\frac{\ln a+\ln b}{2}\right)$, then $\frac{a}{b}+\frac{b}{a}$ is equal to :

• A. $1$
• B. $3$
• C. $5$
• D. $7$

Answer 1

The correct option is D $7$

$\ln \left(\frac{a+b}{3}\right)=\left(\frac{\ln a+\ln b}{2}\right)$

$2\ln \left(\frac{a+b}{3}\right)=\ln \left(ab\right)$

$\ln {\left(\frac{a+b}{3}\right)}^2=\ln \left(ab\right)$

$\therefore {(a+b)}^2=9ab$

$a^2+2ab+b^2=9ab$
$a^2+b^2=7ab$

Dividing throughout by $ab$, we get
$\therefore \frac{a}{b}+\frac{b}{a}=7$