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    Equation of closed curve in x-y plane and its parametric form

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    The equation 49x^2 + 36y^2 = 1764 describes a certain closed curve in 2-dimensional space. (The caret (^) indicates that the variable to the left of the caret is to be raised to the power to the right of the caret. That is, "x^2" means "x squared" and "y^2" means "y squared".)

    Determine the type of closed curve this equation describes, and express the x and y coordinates of the points on this curve as functions of a single (common) parameter.

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    Solution Preview

    The equation we are given is 49x^2 + 36y^2 = 1764.

    If we divide both sides of this equation by 49, we obtain

    x^2 + (36/49)y^2 = 1764/49

    Now 1764/49 = 36, so we can rewrite this equation as

    x^2 + (36/49)y^2 = 36

    Dividing both sides of this new equation by 36 yields

    (x^2)/36 + (y^2)/49 = ...

    Solution Summary

    The given equation is analyzed for different (but equivalent) forms of it that shed light on the type of curve it describes. Also, a general property of real numbers is used to show how the x and y coordinates of the points on the curve can be expressed in terms of a single (common) parameter. A complete, detailed explanation of all the steps is provided.