Question

Identify the quadric surface. $$ x^{2}-y+z^{2}=0 $$

Short answer

The given equation represents a parabolic cylinder opening upwards along the y-axis.

Step-by-step solution

Rearrange the terms

First, we rearrange the given equation \(x^{2}-y+z^{2}=0 \) to match the standard form of a quadric surface. This gives us \( x^{2} + z^{2} = y \)

Identify the quadric surface

The equation now looks like the general form of a parabolic cylinder, \(x^2 + z^2 = 4p(y-k) \), where p is the distance from the vertex to the focus, and k is the shift along the y-axis, with \(p=1/4\) and \(k=0\) in our case. It's important to note that a parabolic cylinder always opens in the direction of the isolated variable.