Question

In Exercises \(1-8,\) describe and sketch the surface. $$ y^{2}-z^{2}=4 $$

Short answer

The given equation describes a two-sheeted hyperboloid that extends indefinitely along the y-axis. It can be visualized as a hyperbola that has been rotated around the x-axis.

Step-by-step solution

Identify the Type of Surface

First, it's necessary to recognize the type of surface based on the given equation \(y^{2}-z^{2}=4\). The equation can be rearranged to \(y^{2}-z^{2}=2^{2}\). It's in the standard form of a hyperboloid, but only has two variables present, which suggests it might be a hyperbola rotated around the x-axis.

Describe the Surface

The surface is a two-sheeted hyperboloid because it has a negative sign. The hyperboloid here is stretching along the y-axis and curves around the z-axis. Therefore, each fixed value of x determines a rectangular hyperbola.

Sketch the Surface

Begin by graphing the 2D equation \(y^{2}-z^{2}=4\), which forms a rectangular hyperbola. Next, conceptually 'rotate' this hyperbola around the x-axis to create a 3D representation. This creates a two-sheeted hyperboloid that extends indefinitely along the y-axis.