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?
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?
No. I get:
$\displaystyle \frac{x^2}{5}+\frac{y^2}{5/2}+\frac{z^2}{1/2}=1$
That's an ellipsoid that crosses over the axes at $\displaystyle x\pm \sqrt{5}$, $\displaystyle y=\pm\sqrt{5/2}$ and more importantly, at $\displaystyle z=\pm \sqrt{1/2}$ which means the ellipsoid has a z-height of $\displaystyle 1/\sqrt{2}$. 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 $\displaystyle x^2+2y^2=-35$ has no real solution.