The ellipse x^2 + 4y^2 = 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0), then what is the equation of the ellipse?

The given ellipse is x^2/4 + y^2/1 = 1 i.e., the point A, the corner of the rectangle in 1st quadrant, is (2, 1). Again the ellipse circumscribing the rectangle passess through the point (4, 0), so its equation is

x^2/16+y^2/b^2=1

A(2,1) lies on the above ellipse.

4/16+1/b^2=1

1/b^2=1-1/4=3/4

b^2=4/3

Hence the equation of the desired ellipse is x^2/16+3/4 * y^2 = 1

x^2+12y^2=16