Answer
$f(x,y)$ is homogeneous of degree zero.
$$f(x,y)=\frac{\sqrt{3 +5 V^2}}{2 +5 V}$$
Work Step by Step
Given
$$f(x, y)=\frac{\sqrt{3 x^{2}+5 y^{2}}}{2 x+5 y}$$
Let
\begin{aligned}
f (t x, ty)&=\frac{\sqrt{3(t x)^{2}+5(t y)^{2}}}{2 t x+5 t y}\\
&=\frac{t \sqrt{3 x^{2}+5 y^{2}}}{t(2 x+5 y)}\\
&=\frac{\sqrt{3 x^{2}+5 y^{2}}}{2 x+5 y}\\
&=f(x,t)
\end{aligned}
Thus, $f(x,y)$ is homogeneous of degree zero.
Transforming the given function
\begin{aligned}f(x, y)&=\frac{\sqrt{3 \frac{x^{2}}{x^2}+5\frac{y^2}{ x^{2}}}}{2 \frac{x}{x}+5 \frac{y}{x}}\\
&=\frac{\sqrt{3 +5\frac{x^{2}}{y^2} }}{2 +5 \frac{y}{x}}\\
\\
\text{Put} \ \ \ V=\frac{y}{x} \ \ \ \text{so, we get}\\
f(x,y)&=\frac{\sqrt{3 +5 V^2}}{2 +5 V}\\
\end{aligned}
