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Math Help - True or False traces of functions

  1. #1
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    True or False traces of functions

    True of False
    The trace of x^2+2y^2 +10z^2=5 in the plane z=2 is an ellipse. True or False?

    First I substituted z=2 in the equation and got x^2+2y^2=-35. I say False because x^2+2y^2=-35 is not an ellipse. But what is it if it is not any ellipse?
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  2. #2
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    You can put it into the form \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 which is an ellipsoid right. What then do the values a, b, c represent and where is z=2 in all of that?
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  3. #3
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    so a b c are the lengths in the x y z directions. and when z=2 it will be a trace of an ellipse?
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  4. #4
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    No. I get:

    \frac{x^2}{5}+\frac{y^2}{5/2}+\frac{z^2}{1/2}=1

    That's an ellipsoid that crosses over the axes at x\pm \sqrt{5}, y=\pm\sqrt{5/2} and more importantly, at z=\pm \sqrt{1/2} which means the ellipsoid has a z-height of 1/\sqrt{2}. What then does that mean when we ask, where does the ellipsoid cut through the plane z=2 then? No where right? So false. But you can see that since the equation x^2+2y^2=-35 has no real solution.
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