As part of optimization problem I am trying to solve the following system of equations in $x, y \in \mathbb R$
$$ \left\{ \begin{array} 22x+1+y= \frac a2 x \\ 2y+1+x= \frac a8 y\\ x^2+y^2-9<0\\ \frac{x^2}{4}+\frac{y^2}{16}=1\\ \end{array} \right. $$
where $a \in \mathbb R$.
I think there must be some trick how to solve this, but I couldn't find any, so it would be very helpful if someone gives me a hint.
Hint: Plugging $$y=-1-2x+\frac{a}{2}x$$ in the second equation we get $$\frac{1}{16} \left(a^2 (-x)+20 a x+2 a-48 x-16\right)=0$$ Can you doing further?