Hard algebra problem Given   
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$   
Now, it is necessary to find 
$$\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}=?$$  
Is this possible and how? a,b,c are given constants.   
I think, ? is probably a complicated function $F=F(a,b,c)$

The expression 
$$\frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{z^2}{c^4}$$ isn't constant on the ellipsoid $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$$ Notice that I get different values on the points $(x,y,z) = (a,0,0)$ and $(x,y,z)=(0,b,0)$. 
