Ellipse $4x^2+y^2+ax+by+c=0$ passes through the point $(-1,2)$ and is tangent to the x-axis 
Question:
Find the values of the constants $a$,$b$ and $c$ such that the ellipse $4x^2+y^2+ax+by+c=0$ passes through the point $(-1,2)$ and is tangent to the x-axis

What I've done so far:
Substitute $(-1,2)$ in $4x^2+y^2+ax+by+c=0$: 
$$\begin{align}
&\implies 4+4-a+2b+c=0 \\
&\implies c=a-2b-8 \tag{1}
\end{align}$$
Rewriting the equation of the ellipse to get standard form:
$$\begin{align}
& 4x^2+y^2+ax+by=-c \\[6pt]
\implies& 4\left(x^2+\frac{a}{4}x+\frac{a^2}{64}\right)
+\left(y^2+by+\frac{b^2}{4}\right)=a-2b-8+\frac{a^2}{16}+\frac{b^2}{4} \\[6pt]
\implies& \frac{4\left(x+\frac{a}{8}\right)^2}{a-2b-8+\frac{a^2}{16}+\frac{b^2}{4}}+\frac{\left(y+\frac{b}{2}\right)^2}{a-2b-8+\frac{a^2}{16}+\frac{b^2}{4}}=1
\end{align}$$
Since the ellipse is tangent to the $x$-axis:
$$\begin{align}
&\quad\sqrt{a-2b-8+\frac{a^2}{16}+\frac{b^2}{4}} = \frac{b}{2} \\[6pt]
\implies&\quad a-2b-8+\frac{a^2}{16}+\frac{b^2}{4} = \frac{b^2}{4} \\
\implies&\quad 2b= a-\frac{a^2}{16}-8 \tag{2}
\end{align}$$
From $(1)$ and $(2)$:
$$\quad c= a-\left(a-\frac{a^2}{16}-8\right)- 8 \quad\implies\quad c= \frac{a^2}{16} \tag{3}$$ 
Therefore, the equation of the ellipse becomes
$$\frac{4\left(x+\frac{a}{8}\right)^2}{a-(a-\frac{a^2}{16}-8)-8+\frac{a^2}{16}+\frac{b^2}{4}}+\frac{(y+\frac{b}{2})^2}{a-(a-\frac{a^2}{16}-8)-8+\frac{a^2}{16}+\frac{b^2}{4}}=1$$ 
That is,
$$\frac{\left(x+\frac{a}{8}\right)^2}{b^2/16}+\frac{\left(y+\frac{b}{2}\right)^2}{b^2/4}=1 \tag{4}$$ 

I'm stuck at this point and don't know how to find $a$ (or what else to do in general).
All help is appreciated.

There's not one, but an infinite number of ellipses that satisfy the two conditions that you have listed. You have 3 unknowns ($a$,$b$ and $c$) to solve for, but only 2 conditions given by which you can determine them. So, no unique solution.
Note that the equation of the ellipse can be written as 
$$\frac{(x+\frac{a}{8})^2}{(\frac{1}{2})^2} + \frac{(y+\frac{b}{2})^2}{1^2} = \frac{a^2}{4^2}+\frac{b^2}{2^2}+c \equiv D^2$$.
Some edits: I forgot to divide out by $D^2$ to put it in standard form.
So, 
$$\frac{(x+\frac{a}{8})^2}{(\frac{D}{2})^2} + \frac{(y+\frac{b}{2})^2}{D^2} = 1$$.
This shows that the half axis along the y-direction has length D.  Since the ellipse is tangent to the x-axis, the center of the ellipse $(\frac{-a}{8},\frac{-b}{2}$ must be exactly D above the x-axis.  In other words, $\frac{-b}{2}=D$ or $\frac{b^2}{4}=D^2$.
$$\frac{a^2}{4^2}+\frac{b^2}{2^2}+c = \frac{b^2}{4}$$
This is one equation relating $a$,$b$ and $c$. The other is obtained by plugging $(x,y)=(-1,2)$ in to the equation of the ellipse to get the second equation
$$\frac{(x+\frac{a}{8})^2}{(\frac{1}{2})^2} + \frac{(y+\frac{b}{2})^2}{1^2} =  D^2 = \frac{b^2}{4}$$.
Any combination of $a$, $b$ and $c$ that satisfy the two equations above, will give a valid ellipse that meets the two conditions you are given.
