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find the equation in standard form for an ellipse whose Foci is (0,3) and (0,-3) with a major axis, length 10

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The center of the ellipse is at the midpoint between the foci, which is the origin (0, 0). The distance from the center to each focus is 3, which is half the length of the major axis. Therefore, the length of the semi-major axis is 5.

The standard form of the equation of an ellipse with center (h, k), semi-major axis a, and semi-minor axis b is:

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

Plugging in the values we know, we get:

(x - 0)^2 / 5^2 + (y - 0)^2 / b^2 = 1

Simplifying, we get:

x^2 / 25 + y^2 / b^2 = 1

To find the value of b, we can use the formula:

c^2 = a^2 - b^2

where c is the distance from the center to each focus, which is 3. Plugging in the values we know, we get:

3^2 = 5^2 - b^2

Simplifying, we get:

b^2 = 16

Therefore, the equation of the ellipse in standard form is:

x^2 / 25 + y^2 / 16 = 1
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