Quadric Surfaces — Question 10

Question 10 – Solution

Strategy Divide by $|\lambda|$ and inspect which side has the single positive term.

$\lambda>0$ The form $x^2/\lambda+y^2/\lambda-z^2/\lambda=1$ is a hyperboloid of one sheet, connected, with axis $z$.

$\lambda=0$ The equation $x^2+y^2=z^2$ is a double circular cone with vertex at the origin.

$\lambda<0$ Writing $z^2/|\lambda|-x^2/|\lambda|-y^2/|\lambda|=1$ gives a hyperboloid of two sheets. Its vertices are $\boxed{(0,0,\pm\sqrt{|\lambda|})}$.

Transition At zero, the one-sheet surface pinches to a cone and separates into two components.