/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 Determine the intersection of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 51

Determine the intersection of the hyperbolic paraboloid \(z=y^{2} / b^{2}-x^{2} / a^{2}\) with the plane \(b x+a y-z=0\). (Assume \(a, b>0 .)\)

Short Answer

Expert verified
The intersection lines can be found by setting the hyperbolic paraboloid equation equal to the plane equation and solve for \(y\) and \(x\). After substituting and simplifying the equations, final values for \(x\) and \(y\) could be obtained.

Step by step solution

01

Express the hyperbolic paraboloid equation and the plane equation

The hyperbolic paraboloid is given by \(z=y^{2}/b^{2}-x^{2}/a^{2}\) and plane by \(bx+ay-z=0\). Our task is to find the intersection by setting both equations equal to each other and solving for \(x\) and \(y\).
02

Equating the equations

Setting the values of \(z\) from both equations to be equal, we obtain: \(y^{2}/b^{2}-x^{2}/a^{2} = bx+ay\). It is now in a form to solve for \(x\) and \(y\).
03

Isolate one variable and solve

Perform a manipulation to isolate any one variable, let's choose \(y\). So, \(y^{2}/b^{2} = x^{2}/a^{2} + bx+ay\). From here, solve for \(y\).
04

Find the value of x

Substitute the value of \(y\) from step 3 into the plane equation \(-bx+ay-z=0\) and solve for \(x\).
05

Final simplification

Use the results that have been obtained to simplify the equations and find the intersection.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(25-28,\) find the direction cosines of \(u\) and demonstrate that the sum of the squares of the direction cosines is 1. $$ \mathbf{u}=\mathbf{i}+2 \mathbf{j}+2 \mathbf{k} $$

Find \((\mathbf{a}) \mathbf{u} \cdot \mathbf{v},(\mathbf{b}) \mathbf{u} \cdot \mathbf{u},(\mathbf{c})\|\mathbf{u}\|^{2},(\mathbf{d})(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\) and \((e) u \cdot(2 v)\). $$ \mathbf{u}=\mathbf{i}, \quad \mathbf{v}=\mathbf{i} $$

Let \(\mathbf{r}=\langle x, y, z\rangle\) and \(\mathbf{r}_{0}=\langle 1,1,1\rangle .\) Describe the set of all points \((x, y, z)\) such that \(\left\|\mathbf{r}-\mathbf{r}_{0}\right\|=2\)

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=-\frac{1}{3}(\mathbf{i}-2 \mathbf{j}) \\ \mathbf{v}=2 \mathbf{i}-4 \mathbf{j} \end{array} $$

The vector \(\mathbf{u}=\langle 3240,1450,2235\rangle\) gives the numbers of hamburgers, chicken sandwiches, and cheeseburgers, respectively, sold at a fast-food restaurant in one week. The vector \(\mathbf{v}=\langle 1.35,2.65,1.85\rangle\) gives the prices (in dollars) per unit for the three food items. Find the dot product \(\mathbf{u} \cdot \mathbf{v},\) and explain what information it gives.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.