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Math Help - parameterize curve of intersection

  1. #1
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    parameterize curve of intersection

    Ellipsoid has equation x^2+y^2-yz+z^2=1
    It fits inside a cylinder, which if I did the math correctly, is x^2+y^2-\frac{y^2}{2}+\frac{y^2}{4}=1 Parameterize the curve of intersection of the ellipsoid with the elliptic cylinder.

    So I'm looking for a function that describes where the ellipsoid touches the cylinder?

    I'm not sure where to start
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  2. #2
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    The equations we are given are:

    \begin{aligned}<br />
x^2+y^2-yz+z^2&=1\\<br />
x^2+\frac{3}{4}y^2&=1.<br />
\end{aligned}

    We must parametrize all solutions of this equation as a set of points (x(t),y(t),z(t)). Hint: What happens when the second equation is subtracted from the first?
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  3. #3
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    ok I get \frac{1}{4}y^2-yz+z^2=0
    now I believe the next step is to solve for y or z. I am stuck on how to do this. I used a CAS and got z=y/2
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  4. #4
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    Quote Originally Posted by superdude View Post
    ok I get \frac{1}{4}y^2-yz+z^2=0
    now I believe the next step is to solve for y or z. I am stuck on how to do this. I used a CAS and got z=y/2
    to solve this by hand first multiply the equation by 4 to get

    y^2-4zy=-4z^2 Now lets complete the square on the left hand side

    y^2-4xy+4z^2=-4z^2+4z^2 \iff (y-2z)^2=0 \iff y=2z
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