Explanation Calculation: Consider the provided equation x 2 + 2 x y + y 2 + 4 2 x − 4 2 y = 0 . Note that, if the equation A x 2 + B x y + C y 2 + D x + E y + F = 0 is such that B ≠ 0 , then the x y -term in this equation can be eliminated by a rotation of axes through an angle θ satisfying cot 2 θ = A − C B . Compare the given equation with the standard equation. So, A = 1 , B = 2 , and C = 1 . Substitute the above values in this expression cot 2 θ = A − C B . cot 2 θ = 1 − 1 2 = 0 2 = 0 As cot 2 θ = 0 , when 2 θ = π 2 . Thus, the value of θ = π 4 . Note that x and y may be obtained from x ′ , y ′ , and θ using the rotation equations x = x ′ c o s θ − y ′ s i n θ and y = x ′ sin θ + y ′ cos θ . Substitute the values of c o s π 4 = 1 2 and sin π 4 = 1 2 in the above equations. x = x ′ c o s θ − y ′ s i n θ = x ' 1 2 − y ′ 1 2 = x ' − y ′ 2 And, y = x ′ sin θ + y ′ cos θ = x ′ 1 2 + y ′ 1 2 = x ′ + y ′ 2 Substitute the above calculated values of x and y in the provided equation. x ′ − y ′ 2 2 + 2 x ′ − y ′ 2 x ′ + y ′ 2 + x ′ + y ′ 2 2 + 4 2 x ′ − y ′ 2 − 4 2 x ′ + y ′ 2 = 0 1 2 x ′ 2 − 2 x ′ y ′ + y ′ 2 + 2 2 x ′ 2 − y ′ 2 + 1 2 x ′ 2 + 2 x ′ y ′ + y ′

Calculus Early Transcendentals, Binder Ready Version

11th Edition

ISBN: 9781118883822

Author: Howard Anton, Irl C. Bivens, Stephen Davis

Publisher: WILEY

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Chapter 10.5, Problem 19ES

To determine

To calculate: the vertex, focus, and directrix after showing the provided equation represents a parabola.

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Chapter 10.5, Problem 18ES

Chapter 10.5, Problem 20ES

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Chapter 10 Solutions

Calculus Early Transcendentals, Binder Ready Version

Ch. 10.1 - Prob. 1QCE Ch. 10.1 - The graph of the curve described by the parametric...Ch. 10.1 - Suppose that a parametric curve C is given by the...Ch. 10.1 - To find dy/dx directly from the parametric...Ch. 10.1 - Let L be the length of the curve x=lnt,y=sint1t An...Ch. 10.1 - (a) By eliminating the parameter, sketch the...Ch. 10.1 - (a) By eliminating the parameter, sketch the...Ch. 10.1 - Sketch the curve by eliminating the parameter, and...Ch. 10.1 - Sketch the curve by eliminating the parameter, and...Ch. 10.1 - Sketch the curve by eliminating the parameter, and...

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the vertex, focus, and directrix after showing the provided equation represents a parabola.

Solution Summary: The author explains that the vertex, focus, and directrix of the provided equation represent a parabola.

To calculate: the vertex, focus, and directrix after showing the provided equation represents a parabola.

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Chapter 10 Solutions