# Thread: True or False traces of functions

1. ## 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?

2. 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?

3. 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?

4. 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.