What is the equation of an ellipse ?

Question

Find the equation of an ellipse that passes through the points (2,3) and (1,-4)

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Answer 1 · 2016-12-24T08:57:55

#(y - -4)^2/7^2 + (x - 2)^2/1^2 = 1#

Excluding rotations, there are two general Cartesian forms for the equation of an ellipse:

#(x - h)^2/a^2 + (y - k)^2/b^2 = 1" [1]"#

and:

#(y - k)^2/a^2 + (x - h)^2/b^2 = 1" [2]"#

let k = -4, h = 2, and use equation [2}:

#(y - -4)^2/a^2 + (x - 2)^2/b^2 = 1#

Use the point #(2,3)#

#(3 - -4)^2/a^2 + (2 - 2)^2/b^2 = 1#

#a = 7#

Use the point #(1, -4)#:

#(-4 - -4)^2/a^2 + (1 - 2)^2/b^2 = 1" [2]"#

#b = 1#

The equation is:

#(y - -4)^2/7^2 + (x - 2)^2/1^2 = 1#