PRECALCULUS:GRAPHICAL,...-NASTA ED.
PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
Expert Solution & Answer
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Chapter 8.4, Problem 27E
Solution

To find: the axis of rotation formulas that will transform x2+xy+y2+1=0 to an equation in u,v coordinates.

The axis of rotation formulas are x=u2v2 and y=u2+v2 .

Given information:

The given quadratic equation is x2+xy+y2+1=0 .

Formula used:

  cot2θ=ACB , where 0<θ<π2 for the quadratic equation Ax2+Bxy+Cy2+Dx+Ey+F=0 .

Axis-rotation formulas are x=ucosθ-vsinθ and y=usinθ+vcosθ .

Calculation:

Compare the given equation with quadratic equation Ax2+Bxy+Cy2+Dx+Ey+F=0 find the values of constants, A,B, and C .

  A=1B=1C=1

Substitute the obtained values of A,B, and C into the formula, cot2θ=ACB and solve for angle θ .

  cot2θ=111cot2θ=02θ=cot102θ=π2

Divide the obtained equation by 2 to obtain the value of θ .

  2θ2=π22θ=π4

Substitute π4 for θ into the above formulas to the values the values x and y in u,v -coordinates.

  x=ucosπ4vsinπ4=u12v12=u2v2

  y=usinπ4+vcosπ4=u12+v12=u2+v2

Therefore, the required axis of rotation formulas are x=u2v2 and x=u2+v2 .

Chapter 8 Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

Ch. 8.1 - Prob. 1ECh. 8.1 - Prob. 2ECh. 8.1 - Prob. 3ECh. 8.1 - Prob. 4ECh. 8.1 - Prob. 5ECh. 8.1 - Prob. 6ECh. 8.1 - Prob. 7ECh. 8.1 - Prob. 8ECh. 8.1 - Prob. 9ECh. 8.1 - Prob. 10ECh. 8.1 - Prob. 11ECh. 8.1 - Prob. 12ECh. 8.1 - Prob. 13ECh. 8.1 - Prob. 14ECh. 8.1 - Prob. 15ECh. 8.1 - Prob. 16ECh. 8.1 - Prob. 17ECh. 8.1 - Prob. 18ECh. 8.1 - Prob. 19ECh. 8.1 - Prob. 20ECh. 8.1 - Prob. 21ECh. 8.1 - Prob. 22ECh. 8.1 - Prob. 23ECh. 8.1 - Prob. 24ECh. 8.1 - Prob. 25ECh. 8.1 - Prob. 26ECh. 8.1 - Prob. 27ECh. 8.1 - Prob. 28ECh. 8.1 - Prob. 29ECh. 8.1 - Prob. 30ECh. 8.1 - Prob. 31ECh. 8.1 - Prob. 32ECh. 8.1 - Prob. 33ECh. 8.1 - Prob. 34ECh. 8.1 - Prob. 35ECh. 8.1 - Prob. 36ECh. 8.1 - Prob. 37ECh. 8.1 - Prob. 38ECh. 8.1 - Prob. 39ECh. 8.1 - Prob. 40ECh. 8.1 - Prob. 41ECh. 8.1 - Prob. 42ECh. 8.1 - Prob. 43ECh. 8.1 - Prob. 44ECh. 8.1 - Prob. 45ECh. 8.1 - Prob. 46ECh. 8.1 - Prob. 47ECh. 8.1 - Prob. 48ECh. 8.1 - Prob. 49ECh. 8.1 - Prob. 50ECh. 8.1 - Prob. 51ECh. 8.1 - Prob. 52ECh. 8.1 - Prob. 53ECh. 8.1 - 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