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Chapter 12: Problem 57

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^{-}, \text {and } y z-\text {traces, when they exist. c. Sketch a graph of the surface. $$\frac{y^{2}}{16}+36 z^{2}-\frac{x^{2}}{4}-9=0$$

Short Answer

Expert verified
Based on the given information and the steps taken to analyze and solve the problem, the quadric surface represents a hyperboloid of one sheet with no x-intercept, y-intercepts at (0, 卤12, 0), z-intercepts at (0, 0, 卤0.5), a hyperbola in the xy-plane and xz-plane, and an ellipse in the yz-plane.

Step by step solution

01

Find the intercepts with the coordinate axes

Since there are three axes, we will solve for each axis. 1. x-axis: Set y = 0 and z = 0: $$\frac{0}{16} + 36(0)^2 - \frac{x^2}{4} - 9 = 0$$ $$\frac{x^2}{4} = -9$$ There is no real value of x that would result in x-intercept. 2. y-axis: Set x = 0 and z = 0: $$\frac{y^2}{16} + 36(0)^2 - \frac{0}{4} - 9 = 0$$ $$\frac{y^2}{16} = 9$$ $$y^2 = 144$$ $$y = \pm 12$$ So, y-intercepts are (0, 12, 0) and (0, -12, 0). 3. z-axis: Set x = 0 and y = 0: $$\frac{0}{16} + 36z^2 - \frac{0}{4} - 9 = 0$$ $$36z^2 = 9$$ $$z^2 = \frac{1}{4}$$ $$z = \pm \frac{1}{2}$$ So, z-intercepts are (0, 0, 0.5) and (0, 0, -0.5).
02

Find the equations of xy-, xz-, and yz- traces

1. xy-trace: Set z = 0: $$\frac{y^2}{16} + 36(0)^2 - \frac{x^2}{4} - 9 = 0$$ $$\frac{y^2}{16} - \frac{x^2}{4} = 9$$ This is the equation of an hyperbola in the xy-plane. 2. xz-trace: Set y = 0: $$\frac{0}{16} + 36z^2 - \frac{x^2}{4} - 9 = 0$$ $$36z^2 - \frac{x^2}{4} = 9$$ This is the equation of an hyperbola in the xz-plane. 3. yz-trace: Set x = 0: $$\frac{y^2}{16} + 36z^2 - \frac{0}{4} - 9 = 0$$ $$\frac{y^2}{16} + 36z^2 = 9$$ This is the equation of an ellipse in the yz-plane.
03

Sketch the graph

We've found the following information: Intercepts: - No x-intercept - y-intercepts at (0, 卤12, 0) - z-intercepts at (0, 0, 卤0.5) Traces: - Hyperbola in the xy-plane - Hyperbola in the xz-plane - Ellipse in the yz-plane Now, we can sketch the graph of the quadric surface with the given information. Using a graphing tool or manual drawing, plot the points and traces to see the surface. The surface is a hyperboloid of one sheet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Axis Intercepts
Understanding the coordinate axis intercepts is crucial when studying quadric surfaces. These intercepts are the points where the surface crosses the x, y, and z-axes. To find them, we solve the given equation by setting two variables to zero.

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