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    Description of the curve given in a parametric form

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    2. A curve C has the parametrization x = a sint cos alpha, y = b sint sin alpha ,
    z = c cost , t â?¥ 0 , where a, b , c, alpha are all positive constants.
    a) Show that C lies on the ellipsoid x^2/a^2+y^2/b^2+z^2/c^2=1

    b) Show that C also lies on a plane that contains the z axis.
    c) Describe the curve C. Give its equation.

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    Solution Preview

    We have

    x = alpha*sin(t)*cos(alpha)
    y = b*sin(t)*sin(alpha)
    z = c*cos*t (3)

    (a) Substituting the values of variables given by the above formulas to the left side of the equation of the ellipsoid and simplifying the obtained expression we obtain sin^2t + cos^2t =1.

    This implies that the given curve lies on the given ...

    Solution Summary

    It is determined the shape and the equation of the curve given in a parametric form.