Find the equation of the ellipse given the center (0,0), vertex(0,-4), and focus (0,3)

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The center of the ellipse is (0, 0), one vertex is at (0, -4) and one focus is at (0, 3).

As the vertex and focus lie on the same vertical line the equation of the ellipse is of the form `y^2/a^2 + x^2/b^2 = 1`

The distance between the center and the focus is 3, c = 3. And as the distance between the center and the vertex is 4, a = 4.

`b^2 = a^2 - c^2 = 16 - 9 = 7`

The equation of the ellipse is `y^2/16 + x^2/7 = 1`

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