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Math Help - intersection of the two ellipses

  1. #1
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    Unhappy intersection of the two ellipses

    Find the intersection of the two ellipses:
    \frac{x^2}{9} + \frac{y^2}{4} = 1
    and
    \frac{x^2}{4} + \frac{y^2}{9} = 1

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  2. #2
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    Find the intersection of the two ellipses:
    \frac{x^2}{9} + \frac{y^2}{4} = 1
    and
    \frac{x^2}{4} + \frac{y^2}{9} = 1

    (x^2)/9 +(y^2)/4 = 1 -------------(1)
    (x^2)/4 +(y^2)/9 = 1 -------------(2)

    At the intersection points, the x of one ellipse is the same as the x of the other ellipse. Likewise with the y of both ellipse. So, (1) = (2) at the their intersection.
    (x^2)/9 +(y^2)/4 = (x^2)/4 +(y^2)/9
    (x^2)(1/9 -1/4) = (y^2)(1/9 -1/4)
    x^2 = y^2
    x = y -------------------------------(i)
    Substitute that into, say, (1),
    (x^2)/9 +(x^2)/4 = 1
    (x^2)(1/9 +1/4) = 1
    13x^2 = 36
    x^2 = 36/13
    x = +,-6/sqrt(13)
    Or,
    x = y = +,-(6/13)sqrt(13).

    If you'd draw the two ellipses on the same xy plane, you'd see that there are 4 intersection points. Therefore, the intersection points are:
    ((6/13)sqrt(13),(6/13)sqrt(13)) -----1
    ((6/13)sqrt(13),-(6/13)sqrt(13)) ----2
    (-(6/13)sqrt(13),(6/13)sqrt(13)) ----3
    (-(6/13)sqrt(13),-(6/13)sqrt(13)) ---4
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