Finding lines of symmetry algebraically How would you determine the lines of symmetry of the curve $y=f(x)$ without sketching the curve itself?
I was working on the problem of finding the lines of symmetry of the curve given by:
$ x^{4}$+$ y^{4}$ = $u$ where u is a positive real number.
but I had to resort to doing a quick sketch. 
Is there an explicit condition for a line to be considered a line of symmetry of a curve?
Thanks.

A line L given by $ax+by+c=0$ is a line of symmetry for your curve, if and only if for any point $P_1(x_1,y_1)$ from the curve, its symmetrical point $P_2(x_2,y_2)$ (with respect to the line L) is also on the curve. 
Now:
(1) express $x_2$ and $y_2$ as functions of $x_1, y_1$ (this can be done from the equation of L);
(2) substitute the $x_2$ and $y_2$ expressions in the curve equation (because they should satisfy it too); thus you'll get the "iff condition" which you're looking for. 
In the general case i.e. for any arbitrary curve, it won't be a nice equation, I believe.  
