<s>[INST] Common tangent of two ellipses Find the common tangent to: $4(x-4)^2 +25y^2 = 100$ and $4(x+1)^2 +y^2 = 4$.
I have found the derivatives of the above two equations:
$\dfrac{dy}{dx}=\dfrac{16-4x}{25y}$ and $\dfrac{dy}{dx}=\dfrac{-(4x+4)}{y}$
What do I do next? 
 [/INST] Attempt:
1)$\dfrac{(x-4)^2}{5^2} + \dfrac{y^2}{2^2}=1;$
2) $(x+1)^2 +\dfrac{y^2}{2^2} =1;$
Draw them.
1) Major axis $5$, minor axis $2.$
Centred at $(4,0).$
2) Major axis $2$, minor axis $1.$
Centred at $(-1,0).$
The only common tangents :
1) At $(4,2)$ for ellipse $1.$
2) At $(-1,2)$ for ellipse $2.$
$y=2$;
Check your $dy/dx$ at these points.
Can you find the other common tangent?
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