Given the equation   4x^2 + 3y^2 - 24y = 0 the ellipse has: a)  C = (0,-4); Vert., major axis = 8; foci (0, -6), (0, -2) b)  C = (0, 4); Vert.; major axis = 8; foci (0, 6), (0, 2) c)  C = (0, -4); Horiz.; major axis = 8; foci (2, -4), (-2, -4) d)  C = (0, 4); Horiz.; major axis = 8; foci (2, 4), (-2, 4)

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Given the equation `4x^2 + 3y^2 - 24y = 0`

By completing the square and rewriting in the general form, we can identify center as well as major axis.

Complete the square.

`4x^2 + 3y^2 - 24 y = 0`

`4(x^2) + 3(y^2 - 8y) = 0` Multiply each term by `1/4.`

`(x^2) + 3/4(y^2 - 8y) = 0` Multiply each term by `1/3`

`1/3(x^2) + 1/4(y^2 - 8y) = 0` Complete square for y.

`1/3(x^2) + 1/4(y^2 - 8y + 16) = 0 + (1/4)(16)` Factor.

`x^2/3 + (y - 4)^2/4 = 4` Multiply every term by `1/4`  to get equation =1.

`x^2/12 + (y-4)^2/16 = 1`

From this, we can see that the center is (0, 4).  This now shows us either choice "b" or "d" is correct.  Since a>b, this makes 16 = a^2, therefore the major axis is vertical.  Therefore this eliminates choice "d".

The solution is choice "b".

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