How do you write an equation for a ellipse with center (5, -4), vertical major axis of length 12, and minor axis of length 8?

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Answer 1 · 2016-06-05T02:17:14

#(x - 5)^2/36 + (y + 4)^2/16 = 1#

Standard equations of an ellipse

Major axis is horizontal
#(x - h)^2/a^2 + (y - k)^2/b^2 = 1#

Major axis is vertical
#(x - h)^2/b^2 + (y - k)^2/a^2 = 1#

where:
Center: #(h, k)#
Major axis: #2a#
minor axis: #2b#

In the given
Center: #(5, -4)#
Major axis: #12 = 2a => a = 6#
minor axis: #8 = 2b => b = 4#

We are dealing with an ellipse with a vertical major axis, so we should use the second form of the standard equation

#(x - 5)^2/6^2 + (y + 4)^2/4^2 = 1#

#=> (x - 5)^2/36 + (y + 4)^2/16 = 1#