# How do you determine if f(x,y)=(xy)/sqrt(x^2+y^2) is homogeneous and what would it's degree be?

Feb 19, 2017

$f \left(x\right)$ is homogeneous of degree $1$

#### Explanation:

We have to evaluate:

$f \left(\alpha x , \alpha y\right) = \frac{\left(\alpha x\right) \left(\alpha y\right)}{\sqrt{{\left(\alpha x\right)}^{2} + {\left(\alpha y\right)}^{2}}}$

$f \left(\alpha x , \alpha y\right) = \frac{{\alpha}^{2} x y}{\sqrt{{\alpha}^{2} \left({x}^{2} + {y}^{2}\right)}}$

$f \left(\alpha x , \alpha y\right) = \frac{{\alpha}^{2} x y}{\alpha \sqrt{{x}^{2} + {y}^{2}}}$

$f \left(\alpha x , \alpha y\right) = \frac{\alpha x y}{\sqrt{{x}^{2} + {y}^{2}}}$

So we have:

$f \left(\alpha x , \alpha y\right) = \alpha f \left(x , y\right)$

and we can conclude that $f \left(x\right)$ is homogeneous of degree $1$.