Question

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$-\frac{x^{2}}{3}+3 y^{2}-\frac{z^{2}}{12}=1$$

Short answer

Answer: No, there are no x-intercepts. 2. Are there any z-intercepts for the given surface? Answer: No, there are no z-intercepts. 3. What are the y-intercepts for the given surface? Answer: The y-intercepts are \(y = \pm\sqrt{\frac{1}{3}}\) or \(y = \pm\frac{1}{\sqrt{3}}\). 4. What type of surface is represented by the given equation? Answer: The surface is a hyperboloid of two sheets.

Step-by-step solution

Find the intercepts with the three coordinate axes

To find the intercepts, we set the other two variables to zero and solve the equation for the remaining variable: - For the x-intercept, set \(y = 0\) and \(z = 0\):$$-\frac{x^{2}}{3}+3(0)^{2}-\frac{(0)^{2}}{12}=1$$$$-\frac{x^{2}}{3}=1$$$$x^{2}=-3$$Since it’s impossible to have a negative value for \(x^2\), there is no x-intercept. - For the y-intercept, set \(x = 0\) and \(z = 0\):$$-\frac{(0)^{2}}{3}+3 y^{2}-\frac{(0)^{2}}{12}=1$$$$3 y^{2}=1$$$$y^{2}=\frac{1}{3}$$The y-intercepts are \(y = \pm\sqrt{\frac{1}{3}}\) or \(y = \pm\frac{1}{\sqrt{3}}\). - For the z-intercept, set \(x = 0\) and \(y = 0\):$$-\frac{(0)^{2}}{3}+3(0)^{2}-\frac{z^{2}}{12}=1$$$$-\frac{z^{2}}{12}=1$$$$z^{2}=-12$$Since it’s impossible to have a negative value for \(z^2\), there is no z-intercept.

Find the \(x y-, x z^{-}\), and \(y z\) -traces

To find the traces, set one of the coordinates to zero and solve the resulting equation: - \(xy\)-trace (\(z = 0\)):$$-\frac{x^{2}}{3}+3y^{2}=\frac{z^{2}}{12}$$$$-\frac{x^{2}}{3}+3y^{2}=0$$ - \(xz\)-trace (\(y = 0\)):$$-\frac{x^{2}}{3}+3(0)^{2}-\frac{z^{2}}{12}=1$$$$-\frac{x^{2}}{3}-\frac{z^{2}}{12}=1$$ - \(yz\)-trace (\(x = 0\)):$$-\frac{(0)^{2}}{3}+3y^{2}-\frac{z^{2}}{12}=1$$$$3y^{2}-\frac{z^{2}}{12}=1$$

Sketch a graph of the surface

Now that we have the intercepts and traces, we can combine the information to sketch a graph of the surface. Since there are different online tools available to help us visualize the surface properly, we can use them to create the graph. One such tool to visualize the graph is GeoGebra. Here's a link to a graph of the surface: [Hyperboloid of two sheets](https://www.geogebra.org/3d?lang=en&q=-%5Cfrac%7Bx%5E%7B2%7D%7D%7B3%7D%2B3y%5E%7B2%7D-%5Cfrac%7Bz%5E%7B2%7D%7D%7B12%7D%3D1) This surface is known as a hyperboloid of two sheets. We can observe from the graph that there are no x-intercepts and z-intercepts, but there are two nonzero y-intercepts as we calculated earlier. The \(xy\), \(xz\), and \(yz\) traces are represented in their respective planes in the 3D space.

Key concepts

Coordinate Intercepts

In the context of quadric surfaces like the hyperboloid of two sheets, finding coordinate intercepts involves determining where the surface intersects the coordinate axes. This means setting two of the three variables (x, y, z) to zero and solving for the third variable to find the intercept.

For example, to find the x-intercept, we set both y and z to zero and look for a solution to the resulting equation. In our equation, \[-\frac{x^{2}}{3}+3 y^{2}-\frac{z^{2}}{12}=1\],
setting y = 0 and z = 0 gives us:
\[ -\frac{x^{2}}{3}=1\]
which leads to \(x^{2} = -3\). Since \(x^{2}\) cannot be negative, this means there's no x-intercept.

Repeating this process for y and z, we find:

• **Y-intercept**: Setting x = 0 and z = 0, gives \(y = \pm\frac{1}{\sqrt{3}}\).
• **Z-intercept**: Setting x = 0 and y = 0 results in no z-intercept because \(z^{2} = -12\) is not possible.

Coordinate intercepts help us understand the basic points of contact between the surface and the axes, offering an initial glimpse of the shape and extent of the surface in the 3D space.

Traces of Surfaces

The traces of a surface are basically the intersections of the surface with the planes formed by setting one of the coordinates constant (often zero). They help in visualizing the 3-dimensional surface by examining simpler 2-dimensional cross-sections.

For instance, the trace in the xy-plane is found by setting z = 0 in the hyperboloid equation:
\[-\frac{x^{2}}{3} + 3y^{2} = 0\].
This gives a straight line equation implying that the trace is represented by a single line.

Next, the trace in the xz-plane is found by setting y = 0:
\[-\frac{x^{2}}{3} - \frac{z^{2}}{12} = 1\],
which does not describe a real trace since it results in a positive value (1) on uneven sides.

Lastly, in the yz-plane (x = 0), the resulting equation is:
\[3y^{2} - \frac{z^{2}}{12} = 1\],
providing a hyperbola. These traces are essential tools for sketching graphical representations of the surfaces within their respective planes.

Hyperboloid of Two Sheets

The hyperboloid of two sheets is a specific type of surface in three-dimensional space characterized by its unique geometry. It's recognized by its twin socket-like sheets that are not connected by a central portion. This results in a surface broken into two separate parts, rather distinct from the more familiar single surface.

In mathematical terms, the general equation for a hyperboloid of two sheets is:
\[\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = -1\].
Notice the negative signs in front of two of the squared terms; this key feature differentiates it from other quadric surfaces like ellipsoids and paraboloids.

Given our specific equation, \[-\frac{x^{2}}{3} + 3y^{2} - \frac{z^{2}}{12} = 1\], this indicates a hyperboloid stretching along the y-axis due to the positive term associated with \(y^{2}\). The absence of x-intercepts and z-intercepts highlights how the surface stretches predominantly in the y-direction.

By visualizing this hyperboloid, which typically looks like two distinct, symmetrical bowls or cups facing back to back along the y-axis, students can begin to appreciate the unique and intriguing forms that different equations can represent in three-dimensional space.