Given : $x^{a}=y^{b}=z^{c},y^2=zx$
Let $x^{a} = y^{b} = z^{c}= k$
$\Rightarrow x=k^{\frac{1}{a}}$
$\Rightarrow y=k^{\frac{1}{b}}$
$\Rightarrow z=k^{\frac{1}{c}}$
$\because y^{2}=zx$
$\therefore (k^{\frac{1}{b}})^2$ $=$ $(k^{\frac{1}{c}})*$ $(k^{\frac{1}{b}})$
$\therefore \frac{1}{a}+\frac{1}{b}=\frac{2}{b}$
Option $(C)$ is correct.