8.4 Rotation of axes (Page 4/8)

<< Chapter < Page

College algebra

Page 4 / 8

Chapter >> Page >

Try It

Rewrite the $13 x^{2} - 6 \sqrt{3} x y + 7 y^{2} = 16$ in the $x^{'} y^{'}$ system without the $x^{'} y^{'}$ term.

$\frac{{x^{'}}^{2}}{4} + \frac{{y^{'}}^{2}}{1} = 1$

Got questions? Get instant answers now!

Graphing an equation that has no x′y′ Terms

Graph the following equation relative to the $x^{'} y^{'}$ system:

x^{2} + 12 x y - 4 y^{2} = 30

First, we find $\cot (2 θ) .$

x^{2} + 12 x y - 4 y^{2} = 20 \Rightarrow A = 1, B = 12, and C = −4

\begin{array}{l} \cot (2 θ) = \frac{A - C}{B} \\ \cot (2 θ) = \frac{1 - (−4)}{12} \\ \cot (2 θ) = \frac{5}{12} \end{array}

Because $\cot (2 θ) = \frac{5}{12},$ we can draw a reference triangle as in [link] .

\cot (2 θ) = \frac{5}{12} = \frac{adjacent}{opposite}

Thus, the hypotenuse is

\begin{array}{r} 5^{2} + 12^{2} = h^{2} \\ 25 + 144 = h^{2} \\ 169 = h^{2} \\ h = 13 \end{array}

Next, we find $\sin θ$ and $\cos θ .$ We will use half-angle identities.

\begin{array}{l} \begin{array}{l} \sin θ = \sqrt{\frac{1 - \cos (2 θ)}{2}} = \sqrt{\frac{1 - \frac{5}{13}}{2}} = \sqrt{\frac{\frac{13}{13} - \frac{5}{13}}{2}} = \sqrt{\frac{8}{13} \cdot \frac{1}{2}} = \frac{2}{\sqrt{13}} \end{array} \\ \cos θ = \sqrt{\frac{1 + \cos (2 θ)}{2}} = \sqrt{\frac{1 + \frac{5}{13}}{2}} = \sqrt{\frac{\frac{13}{13} + \frac{5}{13}}{2}} = \sqrt{\frac{18}{13} \cdot \frac{1}{2}} = \frac{3}{\sqrt{13}} \end{array}

Now we find $x$ and $y .$

\begin{array}{l} x = x^{'} \cos θ - y^{'} \sin θ \\ x = x^{'} (\frac{3}{\sqrt{13}}) - y^{'} (\frac{2}{\sqrt{13}}) \\ x = \frac{3 x^{'} - 2 y^{'}}{\sqrt{13}} \end{array}

and

\begin{array}{l} y = x^{'} \sin θ + y^{'} \cos θ \\ y = x^{'} (\frac{2}{\sqrt{13}}) + y^{'} (\frac{3}{\sqrt{13}}) \\ y = \frac{2 x^{'} + 3 y^{'}}{\sqrt{13}} \end{array}

Now we substitute $x = \frac{3 x^{'} - 2 y^{'}}{\sqrt{13}}$ and $y = \frac{2 x^{'} + 3 y^{'}}{\sqrt{13}}$ into $x^{2} + 12 x y - 4 y^{2} = 30.$

\begin{array}{l} {(\frac{3 x^{'} - 2 y^{'}}{\sqrt{13}})}^{2} + 12 (\frac{3 x^{'} - 2 y^{'}}{\sqrt{13}}) (\frac{2 x^{'} + 3 y^{'}}{\sqrt{13}}) - 4 {(\frac{2 x^{'} + 3 y^{'}}{\sqrt{13}})}^{2} = 30 \\ (\frac{1}{13}) [{(3 x^{'} - 2 y^{'})}^{2} + 12 (3 x^{'} - 2 y^{'}) (2 x^{'} + 3 y^{'}) - 4 {(2 x^{'} + 3 y^{'})}^{2}] = 30 & Factor . \\ (\frac{1}{13}) [9 x^{'}^{2} - 12 x^{'} y^{'} + 4 y^{'}^{2} + 12 (6 x^{'}^{2} + 5 x^{'} y^{'} - 6 y^{'}^{2}) - 4 (4 x^{'}^{2} + 12 x^{'} y^{'} + 9 y^{'}^{2})] = 30 & Multiply . \\ (\frac{1}{13}) [9 x^{'}^{2} - 12 x^{'} y^{'} + 4 y^{'}^{2} + 72 x^{'}^{2} + 60 x^{'} y^{'} - 72 y^{'}^{2} - 16 x^{'}^{2} - 48 x^{'} y^{'} - 36 y^{'}^{2}] = 30 & Distribute . \\ (\frac{1}{13}) [65 x^{'}^{2} - 104 y^{'}^{2}] = 30 & Combine like terms . \\ 65 x^{'}^{2} - 104 y^{'}^{2} = 390 & Multiply . \\ \frac{x^{'}^{2}}{6} - \frac{4 y^{'}^{2}}{15} = 1 & Divide by 390 . \end{array}

[link] shows the graph of the hyperbola $\frac{{x^{'}}^{2}}{6} - \frac{4 {y^{'}}^{2}}{15} = 1.$

Got questions? Get instant answers now!

Identifying conics without rotating axes

Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is

A x^{2} + B x y + C y^{2} + D x + E y + F = 0

If we apply the rotation formulas to this equation we get the form

A^{'} {x^{'}}^{2} + B^{'} x^{'} y^{'} + C^{'} {y^{'}}^{2} + D^{'} x^{'} + E^{'} y^{'} + F^{'} = 0

It may be shown that $B^{2} - 4 A C = {B^{'}}^{2} - 4 A^{'} C^{'} .$ The expression does not vary after rotation, so we call the expression invariant . The discriminant, $B^{2} - 4 A C,$ is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.

A General Note

Using the discriminant to identify a conic

If the equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ is transformed by rotating axes into the equation $A^{'} {x^{'}}^{2} + B^{'} x^{'} y^{'} + C^{'} {y^{'}}^{2} + D^{'} x^{'} + E^{'} y^{'} + F^{'} = 0,$ then $B^{2} - 4 A C = {B^{'}}^{2} - 4 A^{'} C^{'} .$

The equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.

If the discriminant, $B^{2} - 4 A C,$ is

$< 0,$ the conic section is an ellipse
$= 0,$ the conic section is a parabola
$> 0,$ the conic section is a hyperbola

Identifying the conic without rotating axes

Identify the conic for each of the following without rotating axes.

$5 x^{2} + 2 \sqrt{3} x y + 2 y^{2} - 5 = 0$
$5 x^{2} + 2 \sqrt{3} x y + 12 y^{2} - 5 = 0$

Let’s begin by determining $A, B,$ and $C .$
$\underset{A}{\underset{︸}{5}} x^{2} + \underset{B}{\underset{︸}{2 \sqrt{3}}} x y + \underset{C}{\underset{︸}{2}} y^{2} - 5 = 0$

Now, we find the discriminant.

$\begin{array}{l} B^{2} - 4 A C = {(2 \sqrt{3})}^{2} - 4 (5) (2) \\ = 4 (3) - 40 \\ = 12 - 40 \\ = - 28 < 0 \end{array}$

Therefore, $5 x^{2} + 2 \sqrt{3} x y + 2 y^{2} - 5 = 0$ represents an ellipse.
Again, let’s begin by determining $A, B,$ and $C .$
$\underset{A}{\underset{︸}{5}} x^{2} + \underset{B}{\underset{︸}{2 \sqrt{3}}} x y + \underset{C}{\underset{︸}{12}} y^{2} - 5 = 0$

Now, we find the discriminant.

$\begin{array}{l} B^{2} - 4 A C = {(2 \sqrt{3})}^{2} - 4 (5) (12) \\ = 4 (3) - 240 \\ = 12 - 240 \\ = - 228 < 0 \end{array}$

Therefore, $5 x^{2} + 2 \sqrt{3} x y + 12 y^{2} - 5 = 0$ represents an ellipse.

Got questions? Get instant answers now!

Questions & Answers

If c is the cost function for a particular product, find the marginal cost functions and their values at x=10 a. c(x) = 800+ 0.04x + 0.0002x² b. c(x) = 250 + 100x + 0.001x²

Mamush Reply

how can I find set theory

Ephraim Reply

how can I find set theory

Jarvis

is there an error on the one about the dime's thickness? says 2.2x10⁶=0.00135 m

Patrick Reply

hi, interested in algebra

Makan Reply

how to reduce an equation?

Makan

by manipulation of both side

9(y+8)-27 is 9y+45. Why can't you reduce that to y+5? I know that's wrong but can't explain why

Patrick Reply

when you reduce an equation to its simplest terms, you can't change the value of the equation. reducing it to y + 5 is equivalent to dividing it by 9 which changes the value. you can multiply it by 1 or 9/9 which would give 9(y + 5). multiplying it by one does not change the value.

Philip

Given a polynomial expression, factor out the greatest common factor.

Hanu Reply

WHAT IS QUADRATIC EQUATION?

Charles Reply

WHAT IS SYSTEM OF LINEAR INEWUALITIES?

Charles

WHAT IS SYSTEM OF LINEAR INEWUALITIES?

Charles

complex perform

Angel

what is equation?

Charles Reply

what are equations?

Charles

Definition of economics according to karl Marx Thomas malthus Jeremy bentham David Ricardo J.K

Rakiya

Please help me is assignment

Rakiya

The 47th problem of Euclid

Kenneth

show that the set of all natural number form semi group under the composition of addition

Nikhil Reply

what is the meaning

Dominic

explain and give four Example hyperbolic function

Lukman Reply

_3_2_1

felecia

⅗ ⅔½

felecia

_½+⅔-¾

felecia

The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu

SABAL Reply

1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3

Pawel

2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31

Pawel

Ifeanyi

on number 2 question How did you got 2x +2

Ifeanyi

combine like terms. x + x + 2 is same as 2x + 2

Pawel

x*x=2

felecia

2+2x=

felecia

×/×+9+6/1

Debbie

Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22

Naagmenkoma

Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?

mariel Reply

Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113

Pawel

how do I set up the problem?

Harshika Reply

what is a solution set?

Harshika

find the subring of gaussian integers?

Rofiqul

hello, I am happy to help!

Shirley Reply

please can go further on polynomials quadratic

Abdullahi

hi mam

Mark

I need quadratic equation link to Alpa Beta

Abdullahi Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 3

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask

	5 Key equations By OpenStax Read Online Course
	3 Writing equations of rotated conics in standard form By OpenStax Read Online Course
	2011 Dynamics CRM By Danielrosenberger Start Quiz
	American Politics MCQ By Nicole Bartels Start Quiz
	8 Physiotherapy MMT Review By Rhodes Start Flashcards
	Job Search Skills MCQ By Mary Matera Start Quiz
	Engineering Communication MCQ By Steve Gibbs Start Quiz
	3 Physiotherapy Flashcards Set 3 By Rhodes Start Flashcards
	Art History ARTH209 20th Century By Rebecca Butterfield Start Quiz
	Principles of Marketing By Dionne Mahaffey Start Quiz
	Deciduous Forest By Hope Percle Start Quiz
	26 Toxicology Gustafson- Biotoxins and Venoms By Brooke Delaney Start Exam