Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

In Problems 31– 42, rotate the axes so that the new equation contains no $x y$ -term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.
$5 x^{2} + 6 x y + 5 y^{2} - 8 = 0$

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

The given equation is $5 x^{2} + 6 x y + 5 y^{2} - 8 = 0$ .

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

Step 2. Finding acute angle

Here $A = 5, B = 6, C = 5$

$c o t 2 θ = \frac{5 - 5}{6} = 0$

Where

$c o t 2 θ = 0 2 θ = 90 θ = 45^{\circ}$

Step 3. Finding the values of x and y

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

and

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

Step 4. Substituting these values into the original equation

$5 x^{2} + 6 x y + 5 y^{2} - 8 = 0$

$\frac{5}{2} {(x' - y')}^{2} + 6 \times \frac{1}{\sqrt{2}} (x' - y') \times \frac{1}{\sqrt{2}} (x' + y') + \frac{5}{2} {(x' - y')}^{2} - 8 = 0 \frac{5}{2} (x'^{2} - 2 x' y' + y'^{2}) + 3 (x'^{2} - y'^{2}) + \frac{5}{2} (x'^{2} + 2 x' y' + y'^{2}) = 8$

Step 5. Simplify the above Equation

$\frac{5}{2} (x'^{2} - 2 x' y' + y'^{2}) + 3 (x'^{2} - y'^{2}) + \frac{5}{2} (x'^{2} + 2 x' y' + y'^{2}) = 8 \frac{5}{2} x'^{2} - 5 x' y' + \frac{5}{2} y'^{2} + 3 x'^{2} - 3 y'^{2} + \frac{5}{2} x'^{2} + 5 x' y' + \frac{5}{2} y'^{2} = 8 5 x'^{2} + 5 y'^{2} + 3 x'^{2} - 3 y'^{2} = 8 8 x'^{2} + 2 y'^{2} = 8 x'^{2} + \frac{1}{4} y'^{2} = 1 \frac{x'^{2}}{1} + \frac{y'^{2}}{4} = 1$

This represent equation of ellipse.

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

In Problems 31– 42, rotate the axes so that the new equation contains no $x y$ -term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.
$5 x^{2} + 6 x y + 5 y^{2} - 8 = 0$

Step by Step Solution

Step 1. Given Information

Step 2. Finding acute angle

Step 3. Finding the values of x and y

Step 4. Substituting these values into the original equation

Step 5. Simplify the above Equation

Step 6. Graphing Equation

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

In Problems 31– 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.5x2 + 6xy + 5y2- 8 = 0

Step by Step Solution

Step 1. Given Information

Step 2. Finding acute angle

Step 3. Finding the values of x and y

Step 4. Substituting these values into the original equation

Step 5. Simplify the above Equation

Step 6. Graphing Equation

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In Problems 31– 42, rotate the axes so that the new equation contains no $x y$ -term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.
$5 x^{2} + 6 x y + 5 y^{2} - 8 = 0$