Note that this looks similar to the equation of an elliptical paraboloid Section 6.1 but \(z\) has the same degree as \(x\) and \(y\text{.}\) Consequently, since \(z\) has the same degree as \(x\) and \(y\text{,}\) the traces would behave linearly instead of quadratically. Structurally, from the traces, two hyperbolas and an ellipse. We can see the traces through the intersection of the planes \(x = 2\text{,}\)\(y = -3\text{,}\) and \(z = 1\text{,}\) as shown in Figure 6.11:
Figure6.10.The traces of the elliptic cone \(z^2 = \frac{x^2}{4} + \frac{y^2}{9}\) in the planes \(x = 2\text{,}\)\(y = -3\text{,}\) and \(z = 1\text{.}\)
Use the buttons to slice the surface \(z^2 = \frac{x^2}{4} + \frac{y^2}{9}\) with planes \(x=c\text{,}\)\(y=c\text{,}\) or \(z=c\text{,}\) and drag the slider to vary \(c\text{.}\) The equation of each cross-section is displayed above the figure. Drag the figure to view it from a different angle, or press the “Rotate” button to spin it automatically.
