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Math Help - coordinates of the vertices of the square

  1. #1
    rcs
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    coordinates of the vertices of the square

    A square is inscribed in the ellipse whose equation is (x^2/ 16) + (y^2/9) = 1. find the coordinates of the vertices of the square and the perimeter and the area of the square.
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    MHF Contributor MarkFL's Avatar
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    Re: coordinates of the vertices of the square

    Hint: Solve the non-linear system:

    \frac{x^2}{16}+\frac{y^2}{9}=1

    y=x

    This will give you two opposing vertices, from which you may determine the remaining two by symmetry.

    From there you should have little difficulty in determining the area and perimeter of the square.
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    Member kalyanram's Avatar
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    Re: coordinates of the vertices of the square

    As ellipse is symmetric about the origin and square being a cyclic quadrilateral that is symmetric about the origin with all side equal the only possibility is as show in the figure.
    Let the side of the square be 2a so the vertices are (a,a), (a,-a), (-a,a) (-a,-a) substituting in the equation of the ellipse we have:
    \frac{a^2}{9} + \frac{a^2}{16} = 1 \implies a = \pm \frac{12}{5} \therefore the vertices are (\frac{12}{5}, \frac{12}{5}), (\frac{12}{5}, -\frac{12}{5}), (-\frac{12}{5}, \frac{12}{5}), (-\frac{12}{5}, -\frac{12}{5}). The rest can be determined.
    Attached Thumbnails Attached Thumbnails coordinates of the vertices of the square-ellsq.png  
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