Precalculus with Limits
Precalculus with Limits
3rd Edition
ISBN: 9781133947202
Author: Ron Larson
Publisher: Cengage Learning
Question
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Chapter 10, Problem 10CT

a

To determine

To find: The number of degrees the axes must be rotated to eliminate the xy -term.

a

Expert Solution
Check Mark

Answer to Problem 10CT

The number of degrees the axes must be rotated to eliminate the xy -term is 45° .

Explanation of Solution

Given information:

The equation of the conic is x2+6xy+y2−6=0 .

Calculation:

Calculate the rotation of axes to eliminate xy− term.

The general second degree equation is

  Ax2+Bxy+Cy2+Dx+Ey+F=0A′(x′)2+C′(y′)2+D′x′+E′y′+F′=0

By rotating the coordinate axes through an angle θ .

  cot2α=A−BC,0°<2α<180°   ...... (1)

Compare equation x2+6xy+y2−6=0 with the general equation.

  A=1,B=−10,C=1,D=0,E=0,F=−6

Calculate the angle of rotation.

  cot2α=A−CBcot2α=1−1−10cot2α=02α=cot−1(0)

  2α=cot−1(0)α=90°2α=45°

Therefore, the number of degrees the axes must be rotated to eliminate the xy -term is 45° .

b

To determine

To graph: The conic from part (a) and confirm the result using the graphing utility.

b

Expert Solution
Check Mark

Answer to Problem 10CT

The conic from part (a) and confirm the result using the graphing utility is shown in Figure-(1).

Explanation of Solution

Given information:

The equation of the conic is x2+6xy+y2−6=0 .

Calculation:

The equation of the conic in the rotated x′y′ coordinate system

  A′=Acos2α+Bcosαsinα+Csin2α   ...... (1)

  C′=Asin2α−Bcosαsinα+Ccos2α   ...... (2)

  D′=Dcosα+Esinα   ...... (3)

  E′=−Dsinα+Ecosα   ...... (4)

  F′=F   ...... (5)

Here, α=45°

Therefore, the values of sin45° and cos45° are as follows.

  sin45°=12,cos45°=12

Calculate the coefficient of A′,C′,D′,E′andF′ using the equation (2) from (6) .

  A′=1(12)2−4(12)(12)+1(12)2=1

Calculate the coefficient of C′ .

  C′=1(12)2−(−4)(12)(12)+1(12)2=3

Calculate the coefficient of D′ .

  D′=0

Calculate the coefficient of E′ .

  E′=0

Calculate the coefficient of F′ .

  F′=−6

Calculate the equation of conic in xy− plane is

  −1(x′)2+3(y′)2−6=0−(x′)2+3(y′)2=6(y′)22−(x′)26=1

This is the equation of a hyperbola that is centered at the origin of a x′y′ coordinate system having transverse axis on x− axis.

  Precalculus with Limits, Chapter 10, Problem 10CT

Figure-(1)

Therefore, the conic from part (a) and confirm the result using the graphing utility is shown in Figure-(1).

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Chapter 10, Problem 9CT
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Chapter 10, Problem 11CT

Chapter 10 Solutions

Precalculus with Limits

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Prob. 60ECh. 10.7 - Prob. 61ECh. 10.7 - Prob. 62ECh. 10.7 - Prob. 63ECh. 10.7 - Prob. 64ECh. 10.7 - Prob. 65ECh. 10.7 - Prob. 66ECh. 10.7 - Prob. 67ECh. 10.7 - Prob. 68ECh. 10.7 - Prob. 69ECh. 10.7 - Prob. 70ECh. 10.7 - Prob. 71ECh. 10.7 - Prob. 72ECh. 10.7 - Prob. 73ECh. 10.7 - Prob. 74ECh. 10.7 - Prob. 75ECh. 10.7 - Prob. 76ECh. 10.7 - Prob. 77ECh. 10.7 - Prob. 78ECh. 10.7 - Prob. 79ECh. 10.7 - Prob. 80ECh. 10.7 - Prob. 81ECh. 10.7 - Prob. 82ECh. 10.7 - Prob. 83ECh. 10.7 - Prob. 84ECh. 10.7 - Prob. 85ECh. 10.7 - Prob. 86ECh. 10.7 - Prob. 87ECh. 10.7 - Prob. 88ECh. 10.7 - Prob. 89ECh. 10.7 - Prob. 90ECh. 10.7 - Prob. 91ECh. 10.7 - Prob. 92ECh. 10.7 - Prob. 93ECh. 10.7 - Prob. 94ECh. 10.7 - Prob. 95ECh. 10.7 - Prob. 96ECh. 10.7 - Prob. 97ECh. 10.7 - Prob. 98ECh. 10.7 - Prob. 99ECh. 10.7 - Prob. 100ECh. 10.7 - Prob. 101ECh. 10.7 - Prob. 102ECh. 10.7 - Prob. 103ECh. 10.7 - Prob. 104ECh. 10.7 - Prob. 105ECh. 10.7 - Prob. 106ECh. 10.7 - Prob. 107ECh. 10.7 - Prob. 108ECh. 10.7 - Prob. 109ECh. 10.7 - Prob. 110ECh. 10.7 - Prob. 111ECh. 10.7 - Prob. 112ECh. 10.7 - Prob. 113ECh. 10.7 - Prob. 114ECh. 10.7 - Prob. 115ECh. 10.7 - Prob. 116ECh. 10.7 - Prob. 117ECh. 10.7 - Prob. 118ECh. 10.7 - Prob. 119ECh. 10.7 - Prob. 120ECh. 10.7 - Prob. 121ECh. 10.7 - Prob. 122ECh. 10.7 - Prob. 123ECh. 10.7 - Prob. 124ECh. 10.7 - Prob. 125ECh. 10.7 - Prob. 126ECh. 10.7 - Prob. 127ECh. 10.7 - Prob. 128ECh. 10.7 - Prob. 129ECh. 10.7 - Prob. 130ECh. 10.7 - Prob. 131ECh. 10.7 - Prob. 132ECh. 10.7 - Prob. 133ECh. 10.7 - Prob. 134ECh. 10.7 - Prob. 135ECh. 10.7 - Prob. 136ECh. 10.8 - Prob. 1ECh. 10.8 - Prob. 2ECh. 10.8 - Prob. 3ECh. 10.8 - Prob. 4ECh. 10.8 - Prob. 5ECh. 10.8 - Prob. 6ECh. 10.8 - Prob. 7ECh. 10.8 - Prob. 8ECh. 10.8 - Prob. 9ECh. 10.8 - Prob. 10ECh. 10.8 - Prob. 11ECh. 10.8 - Prob. 12ECh. 10.8 - Prob. 13ECh. 10.8 - Prob. 14ECh. 10.8 - Prob. 15ECh. 10.8 - Prob. 16ECh. 10.8 - Prob. 17ECh. 10.8 - Prob. 18ECh. 10.8 - Prob. 19ECh. 10.8 - Prob. 20ECh. 10.8 - Prob. 21ECh. 10.8 - Prob. 22ECh. 10.8 - Prob. 23ECh. 10.8 - Prob. 24ECh. 10.8 - Prob. 25ECh. 10.8 - Prob. 26ECh. 10.8 - Prob. 27ECh. 10.8 - Prob. 28ECh. 10.8 - Prob. 29ECh. 10.8 - Prob. 30ECh. 10.8 - Prob. 31ECh. 10.8 - Prob. 32ECh. 10.8 - Prob. 33ECh. 10.8 - Prob. 34ECh. 10.8 - Prob. 35ECh. 10.8 - Prob. 36ECh. 10.8 - Prob. 37ECh. 10.8 - Prob. 38ECh. 10.8 - Prob. 39ECh. 10.8 - Prob. 40ECh. 10.8 - Prob. 41ECh. 10.8 - Prob. 42ECh. 10.8 - Prob. 43ECh. 10.8 - Prob. 44ECh. 10.8 - Prob. 45ECh. 10.8 - Prob. 46ECh. 10.8 - Prob. 47ECh. 10.8 - Prob. 48ECh. 10.8 - Prob. 49ECh. 10.8 - Prob. 50ECh. 10.8 - Prob. 51ECh. 10.8 - Prob. 52ECh. 10.8 - Prob. 53ECh. 10.8 - Prob. 54ECh. 10.8 - Prob. 55ECh. 10.8 - Prob. 56ECh. 10.8 - Prob. 57ECh. 10.8 - Prob. 58ECh. 10.8 - Prob. 59ECh. 10.8 - Prob. 60ECh. 10.8 - Prob. 61ECh. 10.8 - Prob. 62ECh. 10.8 - Prob. 63ECh. 10.8 - Prob. 64ECh. 10.8 - Prob. 65ECh. 10.8 - Prob. 66ECh. 10.8 - Prob. 67ECh. 10.8 - Prob. 68ECh. 10.8 - Prob. 69ECh. 10.8 - Prob. 70ECh. 10.8 - Prob. 71ECh. 10.8 - Prob. 72ECh. 10.8 - Prob. 73ECh. 10.8 - Prob. 74ECh. 10.8 - Prob. 75ECh. 10.8 - Prob. 76ECh. 10.8 - Prob. 77ECh. 10.8 - Prob. 78ECh. 10.8 - Prob. 79ECh. 10.8 - Prob. 80ECh. 10.9 - Prob. 1ECh. 10.9 - Prob. 2ECh. 10.9 - Prob. 3ECh. 10.9 - Prob. 4ECh. 10.9 - Prob. 5ECh. 10.9 - Prob. 6ECh. 10.9 - Prob. 7ECh. 10.9 - Prob. 8ECh. 10.9 - Prob. 9ECh. 10.9 - Prob. 10ECh. 10.9 - Prob. 11ECh. 10.9 - Prob. 12ECh. 10.9 - Prob. 13ECh. 10.9 - Prob. 14ECh. 10.9 - Prob. 15ECh. 10.9 - Prob. 16ECh. 10.9 - Prob. 17ECh. 10.9 - Prob. 18ECh. 10.9 - Prob. 19ECh. 10.9 - Prob. 20ECh. 10.9 - Prob. 21ECh. 10.9 - Prob. 22ECh. 10.9 - Prob. 23ECh. 10.9 - Prob. 24ECh. 10.9 - Prob. 25ECh. 10.9 - Prob. 26ECh. 10.9 - Prob. 27ECh. 10.9 - Prob. 28ECh. 10.9 - Prob. 29ECh. 10.9 - Prob. 30ECh. 10.9 - Prob. 31ECh. 10.9 - Prob. 32ECh. 10.9 - Prob. 33ECh. 10.9 - Prob. 34ECh. 10.9 - Prob. 35ECh. 10.9 - Prob. 36ECh. 10.9 - Prob. 37ECh. 10.9 - Prob. 38ECh. 10.9 - Prob. 39ECh. 10.9 - Prob. 40ECh. 10.9 - Prob. 41ECh. 10.9 - Prob. 42ECh. 10.9 - Prob. 43ECh. 10.9 - Prob. 44ECh. 10.9 - Prob. 45ECh. 10.9 - Prob. 46ECh. 10.9 - Prob. 47ECh. 10.9 - Prob. 48ECh. 10.9 - Prob. 49ECh. 10.9 - Prob. 50ECh. 10.9 - Prob. 51ECh. 10.9 - Prob. 52ECh. 10.9 - Prob. 53ECh. 10.9 - Prob. 54ECh. 10.9 - Prob. 55ECh. 10.9 - Prob. 56ECh. 10.9 - Prob. 57ECh. 10.9 - Prob. 58ECh. 10.9 - Prob. 59ECh. 10.9 - Prob. 60ECh. 10.9 - Prob. 61ECh. 10.9 - Prob. 62ECh. 10.9 - Prob. 63ECh. 10.9 - Prob. 64ECh. 10.9 - Prob. 65ECh. 10.9 - Prob. 66ECh. 10.9 - Prob. 67ECh. 10.9 - Prob. 68ECh. 10.9 - Prob. 69ECh. 10.9 - Prob. 70ECh. 10.9 - Prob. 71ECh. 10.9 - Prob. 72ECh. 10.9 - Prob. 73ECh. 10.9 - Prob. 74ECh. 10.9 - Prob. 75ECh. 10.9 - Prob. 76ECh. 10.9 - Prob. 77ECh. 10.9 - Prob. 78ECh. 10.9 - Prob. 79ECh. 10.9 - Prob. 80ECh. 10 - Prob. 1RECh. 10 - Prob. 2RECh. 10 - Prob. 3RECh. 10 - Prob. 4RECh. 10 - Prob. 5RECh. 10 - Prob. 6RECh. 10 - Prob. 7RECh. 10 - Prob. 8RECh. 10 - Prob. 9RECh. 10 - Prob. 10RECh. 10 - Prob. 11RECh. 10 - Prob. 12RECh. 10 - Prob. 13RECh. 10 - Prob. 14RECh. 10 - Prob. 15RECh. 10 - Prob. 16RECh. 10 - Prob. 17RECh. 10 - Prob. 18RECh. 10 - Prob. 19RECh. 10 - Prob. 20RECh. 10 - Prob. 21RECh. 10 - Prob. 22RECh. 10 - Prob. 23RECh. 10 - Prob. 24RECh. 10 - Prob. 25RECh. 10 - Prob. 26RECh. 10 - Prob. 27RECh. 10 - Prob. 28RECh. 10 - Prob. 29RECh. 10 - Prob. 30RECh. 10 - Prob. 31RECh. 10 - Prob. 32RECh. 10 - Prob. 33RECh. 10 - Prob. 34RECh. 10 - Prob. 35RECh. 10 - Prob. 36RECh. 10 - Prob. 37RECh. 10 - Prob. 38RECh. 10 - Prob. 39RECh. 10 - Prob. 40RECh. 10 - Prob. 41RECh. 10 - Prob. 42RECh. 10 - Prob. 43RECh. 10 - Prob. 44RECh. 10 - Prob. 45RECh. 10 - Prob. 46RECh. 10 - Prob. 47RECh. 10 - Prob. 48RECh. 10 - Prob. 49RECh. 10 - Prob. 50RECh. 10 - Prob. 51RECh. 10 - Prob. 52RECh. 10 - Prob. 53RECh. 10 - Prob. 54RECh. 10 - Prob. 55RECh. 10 - Prob. 56RECh. 10 - Prob. 57RECh. 10 - Prob. 58RECh. 10 - Prob. 59RECh. 10 - Prob. 60RECh. 10 - Prob. 61RECh. 10 - Prob. 62RECh. 10 - Prob. 63RECh. 10 - Prob. 64RECh. 10 - Prob. 65RECh. 10 - Prob. 66RECh. 10 - Prob. 67RECh. 10 - Prob. 68RECh. 10 - Prob. 69RECh. 10 - Prob. 70RECh. 10 - Prob. 71RECh. 10 - Prob. 72RECh. 10 - Prob. 73RECh. 10 - Prob. 74RECh. 10 - Prob. 75RECh. 10 - Prob. 76RECh. 10 - Prob. 77RECh. 10 - Prob. 78RECh. 10 - Prob. 79RECh. 10 - Prob. 80RECh. 10 - Prob. 81RECh. 10 - Prob. 82RECh. 10 - Prob. 83RECh. 10 - Prob. 84RECh. 10 - Prob. 85RECh. 10 - Prob. 86RECh. 10 - Prob. 87RECh. 10 - Prob. 88RECh. 10 - Prob. 89RECh. 10 - Prob. 90RECh. 10 - Prob. 91RECh. 10 - Prob. 92RECh. 10 - Prob. 93RECh. 10 - Prob. 94RECh. 10 - Prob. 95RECh. 10 - Prob. 96RECh. 10 - Prob. 97RECh. 10 - Prob. 98RECh. 10 - Prob. 99RECh. 10 - Prob. 100RECh. 10 - Prob. 101RECh. 10 - Prob. 102RECh. 10 - Prob. 103RECh. 10 - Prob. 104RECh. 10 - Prob. 105RECh. 10 - Prob. 106RECh. 10 - Prob. 107RECh. 10 - Prob. 108RECh. 10 - Prob. 109RECh. 10 - Prob. 110RECh. 10 - Prob. 111RECh. 10 - Prob. 112RECh. 10 - Prob. 113RECh. 10 - Prob. 114RECh. 10 - Prob. 115RECh. 10 - Prob. 116RECh. 10 - Prob. 117RECh. 10 - Prob. 118RECh. 10 - Prob. 119RECh. 10 - Prob. 1CTCh. 10 - Prob. 2CTCh. 10 - Prob. 3CTCh. 10 - Prob. 4CTCh. 10 - Prob. 5CTCh. 10 - Prob. 6CTCh. 10 - Prob. 7CTCh. 10 - Prob. 8CTCh. 10 - Prob. 9CTCh. 10 - Prob. 10CTCh. 10 - Prob. 11CTCh. 10 - Prob. 12CTCh. 10 - Prob. 13CTCh. 10 - Prob. 14CTCh. 10 - Prob. 15CTCh. 10 - Prob. 16CTCh. 10 - Prob. 17CTCh. 10 - Prob. 18CTCh. 10 - Prob. 19CTCh. 10 - Prob. 20CTCh. 10 - Prob. 21CTCh. 10 - Prob. 22CTCh. 10 - Prob. 1PSCh. 10 - Prob. 2PSCh. 10 - Prob. 3PSCh. 10 - Prob. 4PSCh. 10 - Prob. 5PSCh. 10 - Prob. 6PSCh. 10 - Prob. 7PSCh. 10 - Prob. 8PSCh. 10 - Prob. 9PSCh. 10 - Prob. 10PSCh. 10 - Prob. 11PSCh. 10 - Prob. 12PSCh. 10 - Prob. 13PSCh. 10 - Prob. 14PSCh. 10 - Prob. 15PS
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