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Chapter 11: Problem 53

If a sphere $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{a^{2}}=1 $$ of radius \(a\) is compressed in the \(z\) -direction, then the resulting surface, called an oblate spheroid, has an equation of the form $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1 $$ where \(c

Short Answer

Expert verified
XY-plane: circle with radius \( a \); XZ-plane: ellipse with axes \( 2a \) and \( 2c \).

Step by step solution

01

Understand the Original Sphere Equation

The given sphere equation is \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{a^{2}}=1 \). This represents a sphere centered at the origin with radius \( a \). Every point \((x, y, z)\) on its surface satisfies this equation.
02

Recognize the Oblate Spheroid Equation

The oblate spheroid is represented by \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1 \) where \( c < a \). This means the sphere has been compressed along the \( z \)-axis, making \( c \) the smaller radius in the \( z \) direction.
03

Determine the Circular Trace in the XY-plane

To find the trace in the \( xy \)-plane, set \( z = 0 \) in the oblate spheroid equation: \( \frac{x^2}{a^2} + \frac{y^2}{a^2} = 1 \). This simplifies to \( \frac{x^2 + y^2}{a^2} = 1 \), indicating a circle with radius \( a \).
04

Determine the Elliptical Trace in the XZ-plane

To find the trace in the \( xz \)-plane, set \( y = 0 \) in the oblate spheroid equation: \( \frac{x^2}{a^2} + \frac{z^2}{c^2} = 1 \). This is an ellipse. Setting \( x = a \) gives \( z = 0 \) and setting \( z = c \) gives \( x = 0 \). The length of the major axis (along the x-axis) is \( 2a \), and the length of the minor axis (along the z-axis) is \( 2c \).
05

Conclude the Trace Properties

The trace in the \( xy \)-plane is a circle of radius \( a \), and the trace in the \( xz \)-plane is an ellipse with major axis length \( 2a \) and minor axis length \( 2c \). Hence, the oblate spheroid retains these geometric properties when observed in these planes.

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Key Concepts

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

Sphere
A sphere is a perfectly round three-dimensional shape. It's like the 3D version of a circle! All points on a sphere are an equal distance from its center. This distance is known as the radius. In the equation \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{a^{2}}=1 \), each variable \((x, y, z)\) represents coordinates that form the surface of a sphere. The symmetry in this equation shows that the sphere looks the same, no matter how you rotate it. This is why spheres are unique in their uniformity and balance. They have an inherent elegance and simplicity.
Circular Trace
Imagine slicing through our oblate spheroid horizontally at its widest part. When you set \( z = 0 \) in the equation for the oblate spheroid \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1 \), you are left with a simpler formula: \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}=1 \). This is the trace, or outline, in the \( xy \)-plane. What you end up with is a circle with a radius \( a \).

Several interesting points arise:
  • This circle shows the unchanged cross-section in the \( xy \)-plane despite the flattening in the \( z \)-direction.
  • The equation confirms that the circle retains its radius as \( a \), representing symmetry around this horizontal slice.
Elliptical Trace
The elliptical trace is what you see when you look at the oblate spheroid from the side. If you set \( y = 0 \), the oblate spheroid equation simplifies to \( \frac{x^{2}}{a^{2}} + \frac{z^{2}}{c^{2}} = 1 \). This is an ellipse. It's slightly squashed along the \( z \)-axis. Some vital characteristics of this ellipse are:
  • The major axis, which is the longest distance across the ellipse, lies along the \( x \)-axis. It has a length of \( 2a \).
  • The minor axis, which is shorter, lies along the \( z \)-axis, with length \( 2c \). This reflects the compression from the original sphere.
That's why the sphere looks like an ellipse when viewed through the \( xz \)-plane.
XY-plane
The \( xy \)-plane is a flat surface that stretches infinitely in all directions horizontally. It's like a vast floor with no walls or ceiling. In terms of our sphere and oblate spheroid, the \( xy \)-plane is essential to understanding the circular trace.To find the circular trace, setting \( z = 0 \) simplifies our equation and allows us to visualize the cross-section as a top-down view. This plane is crucial because it keeps the radius of the sphere intact, helping to identify if any flattening or distortion happens in other planes.
XZ-plane
The \( xz \)-plane is the vertical flat surface where only the x-axis and z-axis operate. There's no y-axis here, so this plane allows a side view of the shapes we consider. In our case, it gives insight into the vertical flattening of the sphere into the oblate spheroid. When we set \( y = 0 \) in the spheroid's equation, we examine this vertical cross-section. This is where the elliptical trace becomes significant, as we see the compression effects clearly, forming an ellipse with a major axis of \( 2a \) and a minor axis of \( 2c \).

The importance of analyzing the \( xz \)-plane is based on:
  • Showing how shapes change or adapt under applied transformations.
  • Illustrating different aspects of geometry, highlighting structural changes that aren't visible from just one plane.

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Most popular questions from this chapter

Express the vector \(\mathbf{v}\) as the sum of a vector parallel to \(\mathbf{b}\) and a vector orthogonal to \(\mathbf{b}\). (a) \(\mathbf{v}=2 \mathbf{i}-4 \mathbf{j}, \mathbf{b}=\mathbf{i}+\mathbf{j}\) (b) \(\mathbf{v}=3 \mathbf{i}+\mathbf{j}-2 \mathbf{k}, \mathbf{b}=2 \mathbf{i}-\mathbf{k}\) (c) \(\mathbf{v}=4 \mathbf{i}-2 \mathbf{j}+6 \mathbf{k}, \mathbf{b}=-2 \mathbf{i}+\mathbf{j}-3 \mathbf{k}\)

Use the method of slicing to show that the volume of the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$ is \(\frac{4}{3} \pi a b c\).

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Find the distance between the given skew lines. $$ \begin{aligned} &x=3-t, y=4+4 t, z=1+2 t \\ &x=t, y=3, z=2 t \end{aligned} $$

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