What is the equation of the ellipse with foci (0 6) (0 -6) and co-vertices (2 0) (-2 0)?

The foci of the ellipse are at (0, 6) and (0, -6), indicating that the ellipse is vertically oriented. The distance between the foci is 12, so the distance ( c ) is 6 (half of 12). The co-vertices at (2, 0) and (-2, 0) give the length of the semi-major axis ( a = 2 ). The standard equation of the ellipse is given by ( \frac{x^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 ), where ( k = 0 ), ( a^2 = 36 ), and ( b^2 = a^2 - c^2 = 4 - 36 = -32 ). Therefore, the equation is ( \frac{x^2}{4} + \frac{y^2}{36} = 1 ).