二次曲面

Second-degree equations cut the standard surfaces

Hyperboloids and cones come in sheets

Traces are the fastest identification tool

Which quadric surface is described by the equation $z = x^2 + 4y^2$?

Which surface is the set $x^2 + y^2 - z^2 = -1$?

The cone $z^2 = x^2 + y^2$ meets the horizontal plane $z = 3$. What is the trace?

For the hyperboloid of one sheet $x^2 + y^2 - z^2 = 1$, the horizontal slice at height $z$ is a circle of radius $\sqrt{1 + z^2}$. Give the radius of the narrowest slice, as a plain number.

The surface $\dfrac{(x-1)^2}{4} + y^2 + (z+2)^2 = 1$ is an ellipsoid centered at a point $(a, b, c)$. Give the value of $a$, the first coordinate of the center, as a plain number.