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Chapter 1: Problem 54

Hyperboloid of one sheet \(25 x^{2}+25 y^{2}-z^{2}=25\) and elliptic cone \(-25 x^{2}+75 y^{2}+z^{2}=0\) are represented in the following figure along with their intersec tion curves. Identify the intersection curves and find their equations (Hint: Find \(y\) from the system consisting of the equations of the surfaces.)

Short Answer

Expert verified
The intersection curves are circles: \(y = \pm \frac{1}{2}, x^{2} + z^{2} = \frac{5}{2}\).

Step by step solution

01

Recognize the Forms

First, observe the equations given: 1. The hyperboloid of one sheet: \( 25x^{2}+25y^{2}-z^{2}=25 \)2. The elliptic cone: \( -25x^{2}+75y^{2}+z^{2}=0 \)These need to be manipulated to find the curves of intersection.
02

Simplify Equations

Rearrange both equations by dividing through by their respective constants:1. Hyperboloid: \( x^{2} + y^{2} - \frac{z^{2}}{25} = 1 \)2. Elliptic Cone: \( -x^{2} + 3y^{2} + \frac{z^{2}}{25} = 0 \)
03

Solve for z in Both Equations

From the elliptic cone equation, solve for \(z^{2}\):\[ z^{2} = 25(x^{2} - 3y^{2}) \]Substitute in the hyperboloid equation:\[ x^{2} + y^{2} - (x^{2} - 3y^{2}) = 1 \]
04

Simplify and Solve for y

Combining the equations:\[ x^{2} + y^{2} - x^{2} + 3y^{2} = 1 \]Simplify to find:\[ 4y^{2} = 1 \]Solving gives:\[ y = \pm \frac{1}{2} \]
05

Plug y Back into the Original Equations

Use the values of \( y \) found to get \( x \) and \( z \):For \( y = \frac{1}{2} \, or \, -\frac{1}{2} \), substitute back.1. Substituting into the hyperboloid: \( x^{2} + \left( \frac{1}{2} \right)^{2} - \frac{z^{2}}{25} = 1 \) \( x^{2} + \frac{1}{4} - \frac{z^{2}}{25} = 1 \)2. Substituting into the elliptic cone: \( -x^{2} + 3\left( \frac{1}{2} \right)^{2} + \frac{z^{2}}{25} = 0 \) \( -x^{2} + \frac{3}{4} + \frac{z^{2}}{25} = 0 \)
06

Solve for x and z

Solve these two equations to find: 1. For \( y = \frac{1}{2} \):- These simplify to a circle with \( x^{2} + z^{2} = \frac{5}{2} \)2. For \( y = -\frac{1}{2} \):- Also circular with \( x^{2} + z^{2} = \frac{5}{2} \)
07

Final Intersection Equations

The intersection curves are circles with equations:1. When \( y = \frac{1}{2} \): \( x^{2} + z^{2} = \frac{5}{2} \)2. When \( y = -\frac{1}{2} \): \( x^{2} + z^{2} = \frac{5}{2} \)

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Key Concepts

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

Hyperboloid of One Sheet
The hyperboloid of one sheet is a fascinating type of surface in three-dimensional geometry. It is one of the two types of hyperboloids and can be visualized as a surface that looks somewhat similar to an hourglass or a nuclear cooling tower.
A hyperboloid of one sheet can be represented by the equation: \( 25x^{2}+25y^{2}-z^{2}=25 \). Here, notice the positive signs in front of the squared terms for both \(x\) and \(y\), indicating that these terms contribute to the surface's form in all directions around the axis.
  • The equation can be simplified by dividing through by 25, resulting in the more familiar form \( x^{2} + y^{2} - \frac{z^{2}}{25} = 1 \).
  • This equation is derived from setting the cross-sections of the surface parallel to a particular plane, showing that they are ellipses.
Studying hyperboloids of one sheet offers a glimpse into the descriptive geometry of surfaces, revealing how they can intersect with other forms like cones.
Elliptic Cone
An elliptic cone is another fascinating surface in three-dimensional space. It typically extends infinitely in both directions, forming a shape known for its conical appearance. The elliptic cone combines elements of circles and ellipses to produce its unique shape.
The equation for an elliptic cone might look like this: \( -25x^{2}+75y^{2}+z^{2}=0 \). Notice here that one of the terms is negative: this is essential to understanding the cone's shape as it passes through the origin.
  • By rearranging the original equation and dividing through the coefficients, one can express it as \( -x^{2} + 3y^{2} + \frac{z^{2}}{25} = 0 \).
  • The surface is symmetric with respect to its vertical axis, showcasing the elliptical cross-sections.
Engaging with elliptic cones helps understand complex intersections, especially when combined with other geometric figures like hyperboloids.
Intersection Curves
Intersection curves occur when two surfaces in space cross each other, creating unique shapes dependent on their equations. For this particular exercise, we are interested in the interaction between a hyperboloid of one sheet and an elliptic cone.
These intersection curves help us visualize geometric properties such as circles formed by intersecting planes or sections of these three-dimensional objects.
  • In our problem, solving and equating both original equations leads us to recognize how these surfaces intersect.
  • By systematically solving, we found the points of intersection to be circles described by planes at specific values of \(y\) (i.e., \( y = \pm \frac{1}{2} \)).
Exploring intersection curves introduces us to crucial concepts of spatial reasoning, showcasing how complex surfaces can connect at certain geometric points and planes.
Equation Solving
Equation solving is essential in understanding how geometrical surfaces interact. These solutions require manipulating the given equations to reveal where these shapes intersect or meet.
The art of equation solving lies in carefully managing each step to derive new equations that simplify understanding complex interactions.
  • In our exercise, solving started by managing each of the original equations corresponding to the surfaces given.
  • We found \( z^2 \) from the equation of the elliptic cone, substituted it back into the hyperboloid equation, and simplified to discover \(y = \pm \frac{1}{2}\).
This gave insights into how the shapes intersected, guiding us towards finding the final forms of the intersection curves. Through mathematics, we discover how solving leads to deeper insights into forms and shapes in space.

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Most popular questions from this chapter

A methane molecule has a carbon atom situated at the origin and four hydrogen atoms located at points \(P(1,1,-1), Q(1,-1,1), R(-1,1,1)\), and \(S(-1,-1,-1)\) (see figure) a. Find the distance between the hydrogen atoms located at \(P\) and \(R\). b. Find the angle between vectors \(\overrightarrow{O S}\) and \(\overrightarrow{O R}\) that connect the carbon atom with the hydrogen atoms located at \(S\) and \(R\), which is also called the bond angle. Express the answer in degrees rounded to two decimal places.

For the following exercises, the equations of two planes are given. Determine whether the planes are parallel, orthogonal, or neither. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. \(x+y+z=0,2 x-y+z-7=0\)

A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) \(A(8,0,0), B(8,18,0), C(0,18,8)\), and \(D(0,0,8)\) (see the following figure). a. Find the general form of the equation of the plane that contains the solar panel by using points \(A, B\), and \(C\), and show that its normal vector is equivalent to \(\overrightarrow{A B} \times \overrightarrow{A D}\). b. Find parametric equations of line \(L_{1}\) that passes through the center of the solar panel and has direction vector \(\mathbf{s}=\frac{1}{\sqrt{3}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}\), which points toward the position of the Sun at a particular time of day. c. Find symmetric equations of line \(L_{2}\) that passes through the center of the solar panel and is perpendicular to it. d. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines \(L_{1}\) and \(L_{2}\).

Rewrite the given equation of the quadric surface in standard form. Identify the surface. $$ 63 x^{2}+7 y^{2}+9 z^{2}-63=0 $$

A spheroid is an ellipsoid with two equal semiaxes. For instance, the equation of a spheroid with the z-axis as its axis of symmetry is given by \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1\), where \(a\) and \(c\) are positive real numbers. The spheroid is called oblate if \(ca\). a. The eye cornea is approximated as a prolate spheroid with an axis that is the eye, where \(a=8.7 \mathrm{~mm}\) and \(c=9.6 \mathrm{~mm} .\) Write the equation of the spheroid that models the cornea and sketch the surface. b. Give two examples of objects with prolate spheroid shapes.
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