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?

1 Answer
Sep 26, 2016

Answer:

#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#


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)#


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#


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#


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#