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Math Help - Two points on an elipse; trying to find foci and vertex/minor axis

  1. #1
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    Two points on an elipse; trying to find foci and vertex/minor axis

    Hi there new here. This seems like a really easy question but can't get it:

    basically standard form of ellipse:

    Code:
    x2     y2
    --  +  --  = 1
    a2     b2
    Here's the question: ''pass through (2,2) and (1,4)''

    plz solve, thanks!

    (what I did: filled in two of these equations with this info on top and two sets of unknown a's and b's, but have no idea where to take it after that.)
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  2. #2
    Super Member bigwave's Avatar
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    maybe I am misunderstanding your question but any number of elipses could pass through 2 given points we would need more info

    besides this. welcome to the forum, it is a great place to be.
    Last edited by bigwave; December 26th 2010 at 10:40 PM. Reason: added welcome to forum
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  3. #3
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    earboth's Avatar
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    Quote Originally Posted by smash View Post
    Hi there new here. This seems like a really easy question but can't get it:

    basically standard form of ellipse:

    Code:
    x2     y2
    --  +  --  = 1
    a2     b2
    Here's the question: ''pass through (2,2) and (1,4)''

    plz solve, thanks!

    (what I did: filled in two of these equations with this info on top and two sets of unknown a's and b's, but have no idea where to take it after that.)
    1. I assume that you already have:

    \left|\begin{array}{rcl}\frac4{a^2}+\frac4{b^2} &=&1 \\ \frac1{a^2}+\frac{16}{b^2} &=&1\end{array}\right. which yields: \left|\begin{array}{rcl}4b^2+4a^2 &=&a^2 b^2 \\ b^2+16a^2&=& a^2 b^2\end{array}\right.

    2. Subtract the 2nd equation from the 1st one and you'll get:

    b^2 = 4a^2~\implies~|b| = 2\cdot |a|

    3. Plug in this value into the 1st (or 2nd) equation. You'll get |a| = \sqrt{5}
    Attached Thumbnails Attached Thumbnails Two points on an elipse; trying to find foci and vertex/minor axis-ellips_zweipkte.png  
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  4. #4
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    Quote Originally Posted by bigwave View Post
    maybe I am misunderstanding your question but any number of elipses could pass through 2 given points we would need more info
    Yes, but the given form, \frac{x^2}{a^2}+ \frac{y^2}{b^2} implies that the axes of symmetry are on the x and y axes and the center is at (0, 0) which reduces it to only one ellipse.

    besides this. welcome to the forum, it is a great place to be.
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