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

Answer: (D)

$7$

$\ln \left(\frac{a+b}{3}\right)=\left(\frac{\ln a+\ln b}{2}\right)$
$\Rightarrow \ln \left(\frac{a+b}{3}\right)=\frac{\ln ab}{2}$
$\Rightarrow \ln \left(\frac{a+b}{3}\right)=\ln \sqrt{ab}$
$\Rightarrow \frac{a+b}{3}=\sqrt{ab}$
On squaring above equation on both sides
$\Rightarrow a^2+b^2+2ab=9ab$
$\Rightarrow \frac{a}{b}+\frac{b}{a}=7$
Ans: D