Q32. In Problems 31鈥� 42, rotate the... [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: Q32. (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.
$x^{2} - 4 x y + y^{2} - 3 = 0$

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

The given equation is $x^{2} - 4 x y + y^{2} - 3 = 0$ .

We need to determine the appropriate rotation formula so that new equation does not contain $x y$ term.

Step 2. Finding acute angle

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

localid="1646901593711" $c o t 2 胃 = \frac{1 - 1}{4} = 0$

Where

$0^{鈭�} < 2 胃 < 180^{鈭�}$

localid="1646901649612" $c o t 2 胃 = 0 2 胃 = 90 胃 = 45^{鈭�}$

Step 3. Finding the value of 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

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

This equation represent hyperbola.

Step 6. Graphing Equation

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding acute angle

Step 3. Finding the value of x and y

Step 4. Substituting these values into the original equation

Step 5. Simplify above expression

Step 6. Graphing Equation

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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.x2-4xy+y2-3=0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding acute angle

Step 3. Finding the value of 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

Mechanics Maths

Pure Maths

Decision Maths

Probability and Statistics

Logic and Functions

Calculus

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