calculus – How to solve $ 0 = frac {(a ^ {2} -2 (x ^ {2} + y ^ {2})) x} {(2 (x ^ {2} + y ^ {2}) + a ^ {2}) y} $ only in terms of?

I find the derivative of $ {y} $ w.r.t. $ {x} $ of $ (x ^ {2} + y ^ {2}) ^ {2} = a ^ {2} (x ^ {2} + y ^ {2}) $ be

$
{y} (x) = frac {(a ^ {2} -2 (x ^ {2} + y ^ {2})) x} {(2 (x ^ {2} + y ^ 2)) + a ^ {2}) y}
$

where y is a function of x and a is a positive constant. The question then asks for points where the original function is parallel to the x-axis. After setting the derivative above to 0, I end with a result that has x in terms of y and a. Solutions give $ x = pm frac {1} {4} a sqrt {6} $. How can you solve this problem only in terms of? Thank you in advance for any help! (apologize for any poor latex code)