Quadric Surfaces — Question 4

Question 4 – Solution

Strategy A homogeneous quadratic with two positive squares and one negative square describes a cone.

Classification The equation is $4x^2+y^2=9z^2$, an elliptic cone with vertex at the origin and axis the $z$-axis.

Traces For $z=k\ne 0$, $4x^2+y^2=9k^2$ is an ellipse; for $z=0$ it is only the vertex. For fixed $x=k$ or $y=k$, the traces are hyperbolas when the fixed value is nonzero and intersecting lines when it is zero.

$y=0$ Then $4x^2=9z^2$, so $\boxed{z=\pm\frac 23x}$.