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Chapter 11: Problem 7

Find equations of the traces in the coordinate planes and sketch the traces in an xyz-coordinate system. [Suggestion: If you have trouble sketching a trace directly in three dimensions, start with a sketch in two dimensions by placing the coordinate plane in the plane of the paper, then transfer the sketch to three dimensions. (a) \(\frac{x^{2}}{9}+\frac{y^{2}}{25}+\frac{z^{2}}{4}=1\) (b) \(z=x^{2}+4 y^{2}\) (c) \(\frac{x^{2}}{9}+\frac{y^{2}}{16}-\frac{z^{2}}{4}=1\)

Short Answer

Expert verified
Find and sketch traces for each equation in 3D by reducing them to 2D plane intersections.

Step by step solution

01

Identify the Problem

The task is to find equations of the traces of the given three-dimensional equations in the coordinate planes (xy, yz, and xz planes) and to sketch these traces in a 3D coordinate system.
02

Analyze Equation (a): Identify Traces

For equation (a) \( \frac{x^{2}}{9}+\frac{y^{2}}{25}+\frac{z^{2}}{4}=1 \), identify the traces by setting each variable to zero. This will give us the intersections (traces) of the given surface with each coordinate plane.
03

Find Traces for Equation (a): xy-plane

Set \( z = 0 \). The equation becomes \( \frac{x^{2}}{9} + \frac{y^{2}}{25} = 1 \). This describes an ellipse centered at the origin with semi-major axis 5 along the y-axis and semi-minor axis 3 along the x-axis.
04

Find Traces for Equation (a): yz-plane

Set \( x = 0 \). The equation becomes \( \frac{y^{2}}{25} + \frac{z^{2}}{4} = 1 \). This describes an ellipse centered at the origin with semi-major axis 5 along the y-axis and semi-minor axis 2 along the z-axis.
05

Find Traces for Equation (a): xz-plane

Set \( y = 0 \). The equation becomes \( \frac{x^{2}}{9} + \frac{z^{2}}{4} = 1 \). This describes an ellipse centered at the origin with semi-major axis 3 along the x-axis and semi-minor axis 2 along the z-axis.
06

Analyze Equation (b): Identify Traces

For equation (b) \( z = x^{2}+4y^{2} \), identify the traces by setting each variable to zero.
07

Find Traces for Equation (b): xy-plane

Since \( z \) is expressed directly as a function of \( x^{2} \) and \( y^{2} \), the trace in the xy-plane is the equation itself: \( z = x^{2}+4y^{2} \). This is a paraboloid opening upwards.
08

Find Traces for Equation (b): yz-plane

Set \( x = 0 \). The equation becomes \( z = 4y^{2} \), a parabolic curve opening upwards along the z-axis.
09

Find Traces for Equation (b): xz-plane

Set \( y = 0 \). The equation becomes \( z = x^{2} \), a parabolic curve opening upwards along the z-axis.
10

Analyze Equation (c): Identify Traces

For equation (c) \( \frac{x^{2}}{9} + \frac{y^{2}}{16} - \frac{z^{2}}{4} = 1 \), identify the traces by setting each variable to zero.
11

Find Traces for Equation (c): xy-plane

Set \( z = 0 \). The equation becomes \( \frac{x^{2}}{9} + \frac{y^{2}}{16} = 1 \). This describes an ellipse centered at the origin with semi-major axis 4 along the y-axis and semi-minor axis 3 along the x-axis.
12

Find Traces for Equation (c): yz-plane

Set \( x = 0 \). The equation becomes \( \frac{y^{2}}{16} - \frac{z^{2}}{4} = 1 \). This is a hyperbola opening along the z-axis.
13

Find Traces for Equation (c): xz-plane

Set \( y = 0 \). The equation becomes \( \frac{x^{2}}{9} - \frac{z^{2}}{4} = 1 \). This is a hyperbola opening along the z-axis.
14

Sketch and Combine 3D Traces

For each equation, sketch the identified traces in their respective 2D planes and then combine them into a 3D view to understand the shape of each surface.

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Key Concepts

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

Traces in Coordinate Planes
In 3D coordinate geometry, traces are the intersections of a surface with a coordinate plane, such as the xy-plane, yz-plane, and xz-plane. When you find the traces of an equation, you set one of the variables (x, y, or z) to zero. This reduces the 3D problem into a 2D problem, making it easier to visualize and sketch. Here's a simple way to think about it:
  • For the xy-plane, set z = 0 and simplify the equation.
  • For the yz-plane, set x = 0 and simplify the equation.
  • For the xz-plane, set y = 0 and simplify the equation.
These simplifications help to visualize the cross-sections of the 3D shape, aiding in understanding its structure.
Ellipsoid
An ellipsoid is a 3D surface that resembles a stretched or squashed sphere. It can be defined by an equation of the form \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]. Each axis length (a, b, and c) represents the semi-major or semi-minor axes.
If an ellipsoid has equal axes (a = b = c), it becomes a sphere. In the given exercise, equation (a) \(\frac{x^2}{9} + \frac{y^2}{25} + \frac{z^2}{4} = 1\) describes an ellipsoid, where:
  • The trace in the xy-plane forms an ellipse with axes 3 and 5.
  • The trace in the yz-plane yields an ellipse with axes 5 and 2.
  • The trace in the xz-plane results in an ellipse with axes 3 and 2.
Each elliptical trace shows how the ellipsoid is stretched in the respective planes. Understanding these traces clarifies how the entire ellipsoid shape is formed.
Paraboloid
A paraboloid looks like a 3D parabola, which can either open upwards or downwards, depending on the equation. The standard forms are:
  • Elliptic paraboloid: \(z = ax^2 + by^2\)
  • Hyperbolic paraboloid: \(z = ax^2 - by^2\)
Given the equation \(z = x^2 + 4y^2\), this represents an elliptic paraboloid opening upwards. The key characteristics are:
  • The trace in the xy-plane is defined by the equation itself \(z = x^2 + 4y^2\), illustrating the full surface of the paraboloid.
  • In the yz-plane, by setting \(x = 0\), the trace simplifies to a parabola \(z = 4y^2\).
  • In the xz-plane, by setting \(y = 0\), the trace simplifies to a different parabola \(z = x^2\).
Each trace reveals how the paraboloid expands in its respective coordinate planes. This helps in visualizing how the surface integrates into 3D space.
Hyperboloid
A hyperboloid is a type of 3D surface that can resemble a double-cone structure or a saddle shape. The standard forms are:
  • Hyperboloid of one sheet: Similar to \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\)
  • Hyperboloid of two sheets: Similar to \(-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)
In the given exercise, equation (c) \(\frac{x^2}{9} + \frac{y^2}{16} - \frac{z^2}{4} = 1\) is a hyperboloid of one sheet. When analyzing its traces:
  • The xz-plane trace, \(\frac{x^2}{9} - \frac{z^2}{4} = 1\), forms a hyperbola opening along the z-axis.
  • The yz-plane trace, \(\frac{y^2}{16} - \frac{z^2}{4} = 1\), also results in a hyperbola.
  • Finally, the xy-plane trace is an ellipse, \(\frac{x^2}{9} + \frac{y^2}{16} = 1\).
Each plane's trace shows how the hyperboloid takes its distinctive 3D shape, with curves opening differently depending on the variable exclusion.

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Most popular questions from this chapter

Find an equation of the surface consisting of all points \(P(x, y, z)\) that are twice as far from the plane \(z=-1\) as from the point \((0,0,1) .\) Identify the surface.

Let \(L_{1}\) and \(L_{2}\) be the lines whose parametric equations are $$ \begin{array}{lll} L_{1}: x=1+2 t, & y=2-t, & z=4-2 t \\ L_{2}: x=9+t, & y=5+3 t, & z=-4-t \end{array} $$ (a) Show that \(L_{1}\) and \(L_{2}\) intersect at the point \((7,-1,-2)\). (b) Find, to the nearest degree, the acute angle between \(L_{1}\) and \(L_{2}\) at their intersection. (c) Find parametric equations for the line that is perpendicular to \(L_{1}\) and \(L_{2}\) and passes through their point of intersection.

$$ x^{2}+y^{2}-6 y=0 $$

Find an equation of the plane that satisfies the stated conditions. The plane through \((1,2,-1)\) that is perpendicular to the line of intersection of the planes \(2 x+y+z=2\) and \(x+2 y+z=3\)

Sketch the surface. $$ z=\sqrt{1-x^{2}-y^{2}} $$
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