Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 5y^2 = 6,$ as shown below. Find $r.$
The equation of the circle is (x - r)^2 + y^2 = r^2
==> 2(x - r) + 2y*y' = 0
The equation of the ellipse is x^2 + 5y^2 = 6
==> 2x + 10y*y' = 0
Matching the derivatives, we get r = 4/5.