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Math Help - Finding area of square in an ellipse

  1. #1
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    Finding area of square in an ellipse

    What is the area of a square inscribed in an ellipse 16x + 9y = 144?

    This is a harder one. I can only work up to the value of a and b but do not know how to find the area of square.

    16x/144 + 9y/144 = 1

    x/9 + y/16 = 1

    So I know a=9, therefore a=3 and b=16 and b=4.

    Because a < b, I know this is a vertical ellipse.

    The next step is to find the area of a square inscribed in the ellipse. But how?

    Thanks.
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  2. #2
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    Quote Originally Posted by shenton View Post
    What is the area of a square inscribed in an ellipse 16x + 9y = 144?

    This is a harder one. I can only work up to the value of a and b but do not know how to find the area of square.

    16x/144 + 9y/144 = 1

    x/9 + y/16 = 1

    So I know a=9, therefore a=3 and b=16 and b=4.

    Because a < b, I know this is a vertical ellipse.

    The next step is to find the area of a square inscribed in the ellipse. But how?

    Thanks.
    The corners of the square will be 4 points (x,y) which satisfy the equation of the ellipse and also x^2 = y^2 since it's a square and thus the sides have equal length.
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  3. #3
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    I'll explain how you actually graph the ellipse becaue the previous poster didn't.

    You have the find the foci

    c^2 = a^2 - b^2

    a= 4 and b=3

    c^2 = 16 - 9
    c= (sqrt)7

    the foci are (0, +/- (sqrt)7)

    the vertices are (0, +/- 4)

    So just graph that and then draw a square inside of the ellipse. Make sure your graph is accurate and then count the sides of the square and multiply the base times the height to get the area.
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  4. #4
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    Quote Originally Posted by shenton View Post
    What is the area of a square inscribed in an ellipse 16x + 9y = 144?...

    The next step is to find the area of a square inscribed in the ellipse. But how?

    Thanks.


    Hello, shenton,

    the vertices of the square lay on the straight lines y = x or y = -x. These lines are the diagonals of the square. (see attachment).

    You only have to calculate the intersects of ellipse and straight line:

    x/9 + y/16 = 1 and y = x ===> x/9 + x/16 = 1 ===> 25/144 x = 1 ===> x = 12/5

    Therefore the 4 vertices of the square are: (12/5, 12/5), (12/5, -12/5),(-12/5, 12/5), (-12/5, -12/5)

    The length of one side of the square is: 12/5 - (-12/5) = 24/5

    Therefore the area of the square is (24/5) = 576/25 = 23.04

    EB



    Attached Thumbnails Attached Thumbnails Finding area of square in an ellipse-quad_in_ellips.gif  
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  5. #5
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    Hello, shenton!

    This is JakeD's solution ... with a diagram.


    What is the area of a square inscribed in an ellipse 16x + 9y = 144?
    Code:
                          |
                         ***
               B    *     |     *    A
            (-p,q)*-------+-------*(p,q)
                 *:       |       :*
                  :       |       :
                * :       |       : *
          - - - * : - - - + - - - : * - - -
                * :       |       : *
                  :       |       :
                 *:       |       :*
                  *-------+-------*(p,-q)
                    *     |     *    C
                         ***
                          |

    We want points A, B, C on the ellipse such that: .AB = AC.
    . . That is: .2p = 2q . . p = q

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . _____
    Solve the equation for y: . y .= .(4/3)√9 - x
    . . . . . . . . . . . . - - . . . . . . . . . . . ._____
    Then p = q becomes: .p .= .(4/3)√9 - p

    . . and we have: .9p .= .16(9 - p) . . x .= .144/25

    Hence: .p = 12/5 = 2.4 .and the side of the square is 4.8 units.


    Therefore, the area of the square is: .4.8 = 23.04 units.

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