The implicit equation

$$(x^2 + 2a y -a^2)^2 = y^2(a^2 - x^2)$$

represents a family of functions the members of the family corresponding to different values of $a$.

Re-arrange the equation into the form $y= f(x)$ and draw some graphs belonging to the family for different values of $a$. What is the domain of the function $f(x)$?

For investigating these graphs you can download the shareware program Graphmatica for free from here as NRICH is an approved distributor of this program. You can find more information about the program from http://www.graphmatica.com/ .