An algebra problem by aaryan vaishya

The question should specify that $x,y,z\neq 0$ (if $y=0$ then $x:z$ could be anything and if $x=z=0$ then $x:z$ is undefined). Assuming that, $\begin{aligned} && \sqrt{xy} &= \sqrt{2yz} \\ &\Rightarrow& \sqrt{\frac{x}{z}} &= \sqrt{2} && \color{#3D99F6} \text{Multiply by }\frac{1}{\sqrt{yz}} \text{ on both sides}\\ &\Rightarrow& \frac{x}{z} &= 2 \end{aligned}$

sqrt x multiplied by sqrt y =sqrt 2 by sqrt z by sqrt y,so sqrt of x/sqrt of z=sqrt of 2 and x/z =2/1