Q31. 聽In Problems 31鈥� 42, rotate t... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 10: Q31. (page 677)

In Problems 31鈥� 42, rotate the axes so that the new equation contains no $x y$ -term. Analyze and graph the new equation. Refer to Problems 21鈥�30 for Problems 31鈥� 40.
localid="1647365908255" $x^{2} + 4 x y + y^{2} - 3 = 0$

Short Answer

Expert verified

$x = \frac{1}{\sqrt{2}} x' - \frac{1}{\sqrt{2}} y' y = \frac{1}{\sqrt{2}} x' + \frac{1}{\sqrt{2}} y'$

Step by step solution

Step 1. Given Information

For the given equation $x^{2} + 4 x y + y^{2} - 3 = 0$ . We need to determine the appropriate rotation so that the new equation does not contain $x y$ term.

Step 1. Finding acute angle

Here $A = 1, B = 4, C = 1$

$c o t 2 胃 = \frac{1 - 1}{4} = 0$

Where

$0^{鈭�} < 2 胃 < 180^{鈭�}$

$c o t 2 胃 = 0 2 胃 = 90 胃 = 45^{鈭�}$

Step 3. Finding x and y.

$x = x' \cos 45^{鈭�} - y' \sin 45^{鈭�} x = \frac{1}{\sqrt{2}} x' - \frac{1}{\sqrt{2}} y'$

and

$y = x' \sin 45^{鈭�} + y' \cos 45^{鈭�} y = \frac{1}{\sqrt{2}} x' + \frac{1}{\sqrt{2}} y'$

Step 4. Substituting these values into the original equation

$x^{2} + 4 x y + y^{2} - 3 = 0$

$\frac{1}{2} {(x' - y')}^{2} + 4 [\frac{1}{\sqrt{2}} (x' - y')] [\frac{1}{\sqrt{2}} (x' + y')] + \frac{1}{2} {(x' + y')}^{2} - 3 = 0 \frac{1}{2} (x'^{2} - 2 x' y' + y'^{2}) + 2 (x'^{2} - y'^{2}) + \frac{1}{2} (x'^{2} + 2 x' y' + y'^{2}) = 3$

Step 5. Simplify above expression

$\frac{1}{2} (x'^{2} - 2 x' y' + y'^{2}) + 2 (x'^{2} - y'^{2}) + \frac{1}{2} (x'^{2} + 2 x' y' + y'^{2}) = 3 \frac{1}{2} x'^{2} - x' y' + \frac{1}{2} y'^{2} + 2 x'^{2} - 2 y'^{2} + \frac{1}{2} x'^{2} + x' y' + \frac{1}{2} y'^{2} = 3 \frac{1}{2} x'^{2} + \frac{1}{2} x'^{2} - x' y' + x' y' + \frac{1}{2} y'^{2} + \frac{1}{2} y'^{2} + 2 x'^{2} - 2 y'^{2} = 3 x'^{2} + y'^{2} + 2 x'^{2} - 2 y'^{2} = 3 3 x'^{2} - y'^{2} = 3 \frac{x'^{2}}{1} - \frac{y'^{2}}{3} = 1$

The equation represent hyperbola.

Step 6. Graphing equation

$\frac{x'^{2}}{1} - \frac{y'^{2}}{3} = 1$

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In Problems 31鈥� 42, rotate the axes so that the new equation contains no $x y$ -term. Analyze and graph the new equation. Refer to Problems 21鈥�30 for Problems 31鈥� 40.
localid="1647365908255" $x^{2} + 4 x y + y^{2} - 3 = 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 1. Finding acute angle

Step 3. Finding x and y.

Step 4. Substituting these values into the original equation

Step 5. Simplify above expression

Step 6. Graphing equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

In Problems 31鈥� 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21鈥�30 for Problems 31鈥� 40.localid="1647365908255" x2+4xy+y2-3=0

Short Answer

Step by step solution

Step 1. Given Information

Step 1. Finding acute angle

Step 3. Finding x and y.

Step 4. Substituting these values into the original equation

Step 5. Simplify above expression

Step 6. Graphing equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 31鈥� 42, rotate the axes so that the new equation contains no $x y$ -term. Analyze and graph the new equation. Refer to Problems 21鈥�30 for Problems 31鈥� 40.
localid="1647365908255" $x^{2} + 4 x y + y^{2} - 3 = 0$