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:

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