multidemensional fixpoint iteration

Question

We want to solve the following system of equation, which shows an intersection between a circle and an ellipse .

$x^2+y^2=5$
$\frac{x^2}{16}+y^2=\frac{5}{4}$

We can expres this system as a fixpoint iteration $(x_k,y_k) = F(x_{k-1},y_{k-1})$ with
$F_1(x,y)=\sqrt{5-y^2}$ and $F_2(x,y) = \sqrt{\frac{5}{4}-\frac{x^2}{16}}$
We define $\omega_x = [0,\sqrt5]$ and $\omega_y=[0,\frac{\sqrt{5}}{2}]$ and $\omega = \omega_x $ x $ \omega_y$

Show that the iteration converges against $x^*=2,y^*=1$ and give a priori estimation how many steps its need, if the exactness is $10^{-6}$. Use the uniform norm $||.||_{\infty}$

My ideas:
First of all i need the two derivations:
$f_1=\frac{-y}{5-y^2}$ and $f_2=-\frac{x}{4\sqrt{20-x^2}}$
The follwing must apply:
$|F'_i|=|f_i|<1$ for $i \in {1,2}$
I get:
$|f_1|<\frac{1}{\sqrt{3}}$ and $|f_2|<\frac{\sqrt3}{12}$
I dont know what to do next. If I try some iterations it seems to be true. For $(x_0,y_0)=(\sqrt{5},\frac{\sqrt{5}}{2})$ you will get $(x_1,y_1)=(0.96825,1.968)$

For the estimation:
$||F'(x,y)||_\infty = max(f_1,f2) < max(\frac{1}{\sqrt{3}},\frac{\sqrt3}{12})=\frac{1}{\sqrt{3}}$
After that i use
$||x_k-x^*|| \leq \frac{L^k}{1-L}||x_1-x_0||$
If i put for $L=\frac{1}{\sqrt{3}}$ and for $||x_k-x^*||=10^{-6}$ i get an imaginary number :(
I hope you can help me