Age 16 to 18 Challenge Level:
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 $f(x)$?
To investigate these graphs you might try http://www.desmos.com.