test of symmetry with respect to the line $y=2x$ how can I check the symmetry of a curve to the line $y=2x$? 
Using geometrical methods I have that the point $({-2\over 5},{8+\sqrt{15}\over 5})$ is the symmetric point of $(2,1)$ with respect to $y=2x$, but I don´t know if this is correct. I´ll appreciate your help.

If you mean by reflection, it is not correct, for  the slope of the line joining any $P$ not on the line to its reflection should be $-\frac{1}{2}$.
Here is one way to solve the problem. Let the point we are looking for be $(a,b)$. 
(i) The line joining $(2,1)$ and $(a,b)$ should be perpendicular to the line $y=2x$. That is, its slope should be $-\frac{1}{2}$.
That will give you a linear equation in $a$ and $b$.
(ii) The midpoint $\left(\frac{a+2}{2},\frac{b+1}{2}\right)$ of the line segment that joins $(1,2)$ and $(a,b)$ should lie on $y=2x$. 
That will give you another linear equation in $a$ and $b$.
Solve the system of two linear equations. I will leave the rest of the setting up, and the calculation, to you. Please indicate if problems arise.
