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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- November 2nd 2009, 02:28 PMandreaspohleTrue 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? - November 2nd 2009, 04:02 PMshawsend
You can put it into the form which is an ellipsoid right. What then do the values a, b, c represent and where is z=2 in all of that?

- November 2nd 2009, 04:21 PMandreaspohle
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?

- November 2nd 2009, 04:54 PMshawsend
No. I get:

That's an ellipsoid that crosses over the axes at , and more importantly, at which means the ellipsoid has a z-height of . 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 has no real solution.