Common tangent of two ellipses

Question

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?

Answer 1

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?