Question

If $a^x=b^y=c^z$ and $b^2=ac$, then $\frac{1}{x}+\frac{1}{z}$ is

• A. $\frac{2}{y}$
• B. $\frac{1}{y}$
• C. $\frac{1}{2y}$
• D. $2y$

Answer 1

The correct option is A $\frac{2}{y}$

Given $a^x=b^y=c^z$ & $b^2=ac$

Let $a^x=b^y=c^z=l$

$xlna=lnl...\left(1\right)$

$ylnb=lnl...\left(2\right)$

$z=lnc=lnl...\left(3\right)$

$2ylnb=2lnl$

$yln{b}^2=2lnl$

$yln\left(ac\right)=2lnl$

$y[lna+lnc]=2lnl$

$y[\frac{lnl}{x}+\frac{lnl}{z}]=2lnl$ (from (1)& (3))

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