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Chapter 12: Problem 48

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. $$4 x^{2}+y^{2}+\frac{z^{2}}{2}=1$$

Short Answer

Expert verified
Based on the step-by-step solution above, provide a short answer to the problem as follows: The given surface is an ellipsoid with semi-principal axes of \(a=\frac{1}{2}\), \(b=1\), and \(c=\sqrt{2}\). It has x-intercepts at \(\left(\frac{1}{2}, 0, 0\right)\) and \(\left(-\frac{1}{2}, 0, 0\right)\), y-intercepts at \(\left(0, 1,0\right)\) and \(\left(0, -1,0\right)\), and z-intercepts at \(\left(0, 0,\sqrt{2}\right)\) and \(\left(0, 0,-\sqrt{2}\right)\). The surface intersects the x y-, x z-, and y z-planes with the equations \(4x^2+y^2=1\), \(4x^2+\frac{z^2}{2}=1\), and \(y^2+\frac{z^2}{2}=1\), respectively.

Step by step solution

01

Finding intercepts with coordinate axes

To find intercepts, we need to solve for x, y, and z when the other two variables are equal to zero. 1. x-intercept: Let y=0 and z=0. Then, the equation becomes: $$4x^2=1 \Rightarrow x^2=\frac{1}{4} \Rightarrow x=\pm\frac{1}{2}$$ So, the x-intercepts are at points \(\left(\frac{1}{2}, 0, 0\right)\) and \(\left(-\frac{1}{2}, 0, 0\right)\). 2. y-intercept: Let x=0 and z=0. Then, the equation becomes: $$y^2=1 \Rightarrow y=\pm 1$$ So, the y-intercepts are at points \(\left(0, 1,0\right)\) and \(\left(0, -1,0\right)\). 3. z-intercept: Let x=0 and y=0. Then, the equation becomes: $$\frac{z^2}{2}=1 \Rightarrow z^2=2 \Rightarrow z=\pm\sqrt{2}$$ So, the z-intercepts are at points \(\left(0, 0,\sqrt{2}\right)\) and \(\left(0, 0,-\sqrt{2}\right)\).
02

Finding equations of x y-, x z-, and y z-traces

Next, we find the equations of the traces when one of the coordinate variables is equal to zero. 1. x y-trace (z=0): The equation becomes: $$4x^2+y^2=1$$ 2. x z-trace (y=0): The equation becomes: $$4x^2+\frac{z^2}{2}=1$$ 3. y z-trace (x=0): The equation becomes: $$y^2+\frac{z^2}{2}=1$$
03

Sketch the graph

Using theinformation gathered from the previous steps, the graph of the surface appears to be an ellipsoid with the semi-principal axes being \(a=\frac{1}{2}\), \(b=1\), and \(c=\sqrt{2}\). The intercepts found in Step 1 can be plotted along the coordinate axes. The surface intersects with the coordinate planes according to the trace equations found in Step 2. We can sketch a graph of the given surface using these traces and the information about the intercepts with the coordinate axes.

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Key Concepts

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

Intercepts
Intercepts are the points where a surface intersects the coordinate axes. In our exercise with the given equation of the quadric surface, we calculate the intercepts by setting two variables to zero and solving for the third. This gives us the points where the surface touches the x, y, and z axes.

Here's how you find them:
  • x-intercept: Set both y and z to zero, solving for x gives us coordinates at \(\left(\frac{1}{2}, 0, 0\right)\) and \(\left(-\frac{1}{2}, 0, 0\right)\).
  • y-intercept: Set both x and z to zero, resulting in y intercepts at \(\left(0, 1, 0\right)\) and \(\left(0, -1, 0\right)\).
  • z-intercept: Set both x and y to zero, giving points at \(\left(0, 0, \sqrt{2}\right)\) and \(\left(0, 0, -\sqrt{2}\right)\).
These intercepts provide key points necessary for sketching the graph of the surface.
Traces
Traces refer to the shapes that are formed when a surface intersects with the coordinate planes. For our quadric surface, traces are found by setting one of the variables x, y, or z to zero and simplifying the equation accordingly. Here's a brief breakdown:

  • xy-trace (when z=0): The equation simplifies to \(4x^2 + y^2 = 1\). This depicts an ellipse in the xy-plane.
  • xz-trace (when y=0): Here, the equation becomes \(4x^2 + \frac{z^2}{2} = 1\), also forming an ellipse, this time in the xz-plane.
  • yz-trace (when x=0): The equation reduces to \(y^2 + \frac{z^2}{2} = 1\), which is another ellipse present in the yz-plane.
Understanding these traces aids in not only graphing the surface but also in visualizing how it interacts with the coordinate planes.
Ellipsoid
An ellipsoid is a type of quadric surface resembling an elongated sphere. It is defined by the general equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\), where a, b, and c are the lengths of the semi-principal axes. Each term with a variable squared ensures the symmetry around the origin.

In our specific case, by rearranging terms and observing intercepts, you can recognize the surface as an ellipsoid. The semi-principal axes can be identified from:
  • a = \(\frac{1}{2}\): derived from setting y and z to zero.
  • b = 1: found by setting x and z to zero.
  • c = \(\sqrt{2}\): from setting x and y to zero.
This ellipsoidal form helps in understanding the 3D shape better and is crucial for sketching the graph accurately.
Graph Sketching
Graph sketching involves plotting the surface by identifying important features such as intercepts, traces, and the nature of the surface. For our given quadric surface, we already know its ellipsoidal nature with the following semi-principal axes: \(a = \frac{1}{2}\), \(b = 1\), and \(c = \sqrt{2}\).

To sketch the graph:
  • Start by plotting the intercepts which are the points where the ellipsoid touches the coordinate axes.
  • Use the trace equations to draw ellipses in the xy, xz, and yz planes. These traces ensure the ellipsoid is accurately centered around the origin.
  • Ensure that the proportions of the axes match their respective values (\(a, b, c\)) to reflect its actual shape and size in 3D space.
By methodically following these steps, you'll successfully visualize how the ellipsoid interacts within the three-dimensional coordinate system, giving you a comprehensive understanding of its geometric properties.

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Most popular questions from this chapter

Find the points (if they exist) at which the following planes and curves intersect. $$\begin{aligned} &2 x+3 y-12 z=0 ; \quad \mathbf{r}(t)=\langle 4 \cos t, 4 \sin t, \cos t\rangle\\\ &\text { for } 0 \leq t \leq 2 \pi \end{aligned}$$

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The temperature of points on an elliptical plate \(x^{2}+y^{2}+x y \leq 1\) is given by \(T(x, y)=25\left(x^{2}+y^{2}\right) .\) Find the hottest and coldest temperatures on the edge of the elliptical plate.

Two resistors in an electrical circuit with resistance \(R_{1}\) and \(R_{2}\) wired in parallel with a constant voltage give an effective resistance of \(R,\) where \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). a. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by solving for \(R\) and differentiating. b. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by differentiating implicitly. c. Describe how an increase in \(R_{1}\) with \(R_{2}\) constant affects \(R\). d. Describe how a decrease in \(R_{2}\) with \(R_{1}\) constant affects \(R\).
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