The ellipse x² + 4y² = 4 is inscribed in a rectangle aligned with the coordinate axes, which is inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is

Given ellipse x² + 4y² = 4
x²/4 + y²/1 = 1…(1)
Given that second ellipse is passing through (4,0)
Substituting (4,0) in x²/a² + y²/b² = 1
4²/a² + 0²/b² = 1
16/a² = 1
a² = 16
The equation of the ellipse is x²/16 + y²/b² = 1
The rectangle inscribed in the ellipse x² + 4y² = 4 has vertices (±2, ±1)
The rectangle is inscribed in another ellipse which passes through (4, 0).
Thus the new ellipse will have a = 4 and b = 2.
The equation of the ellipse is x²/a² + y²/b² = 1
x²/16 + y²/4 = 1
x² + 4y² = 16