Question

How do you write an equation of an ellipse in standard form given foci (0,12) and (0,-12) and a major axis of length 26?

Answer 1

`x^2/25 + y^2/169 = 1`

Explanation:

Given:

`f_1: (0, 12)`

`f_2: (0, -12)`

`M = 2a = 26`

Note that the x-coordinate of the foci are the same. Hence we can conclude that the ellipse has a vertical major axis. The standard equation of an ellipse with a vertical major axis is

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

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Get the center by getting the midpoint between the two foci,

`C_x = (0 + 0)/2`

`=> C_x = 0`

`C_y = (12 + -12)/2`

`=> C_y = 0`

`C: (h, k) => (0, 0)`

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Get `c` by getting the distance between one of the foci and the center. Since we are dealing with an ellipse with a vertical major axis, simply get the difference between the y-coordinates. For this example, let's use the first focus

`c = |(f_1)_y - C_y|`

`c = 12 - 0`

`c = 12`

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Get the value of `b` using the equation

`a^2 - b^2 = c^2`

`13^2 - b^2 = 12^2`

`=> 169 - 144 = b^2`

`=> b^2 = 25`

`=> b = 5`

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Hence, the equation of the ellipse is

`(x - 0)^2/5^2 + (y - 0)^2/13^2 = 1`

`=> x^2/25 + y^2/169 = 1`