Explanation Definition used: Definition 1: “Suppose x and y axes in a coordinate plane are rotated through an acute angle ϕ to produce the X and Y axes. Then the coordinates of a point ( x , y ) in xy -plane in terms of X and Y are given by, x = X cos ϕ − Y sin ϕ and y = X sin ϕ + y cos ϕ ”. Note 1: “The quantity B 2 − 4 A C is invariant under rotation, that is B 2 − 4 A C = B ′ 2 − 4 A ′ C ′ .” Calculation: The given equations is x + y = 1 (1) Convert the equation (1) into standard equation of conic, A x 2 + B x y + c y 2 + D x + E y + F = 0 . Now, equation (1) can be written as follows. x = 1 − y x = ( 1 − y ) 2 x = 1 − 2 y + y x − y − 1 = − 2 y Simplify further, ( x − y − 1 ) 2 = ( − 2 y ) 2 x 2 + y 2 + 1 2 − 2 x y + 2 y − 2 x = 4 y x 2 − 2 x y − y 2 − 2 x + 2 y + 1 = 0 Thus, the equation is x 2 − 2 x y − y 2 − 2 x + 2 y + 1 = 0 (2) It is given that the coordinate axes rotates through an angle 45 ° , by the definition stated above, the xy -coordinates are, x = X cos ϕ − Y sin ϕ = X 2 − Y 2 y = X sin ϕ + Y cos ϕ = X sin 45 ° + Y cos 45 ° = X 2 + Y 2 Substitute the respective values in (1), { 1 2 ( X − Y ) 2 − 2 2 ( X − Y ) 1 2 ( X + Y ) − 3 2

7th Edition

ISBN: 9781305761049

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage

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Question

Chapter 11.5, Problem 36E

To determine

To show: The graph of the equation x+y=1 is part of a parabola by rotating the axes through an angle of 45° .

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Chapter 11.5, Problem 35E

Chapter 11.5, Problem 37E

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Chapter 11 Solutions

Precalculus

Ch. 11.1 - A parabola is the set of all points in the plane...Ch. 11.1 - The graph of the equation x2 = 4py is a parabola...Ch. 11.1 - The graph of the equation y2 = 4px is a parabola...Ch. 11.1 - Label the focus, directrix, and vertex on the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Prob. 6E Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Prob. 8E Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...

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The graph of the equation x + y = 1 is part of a parabola by rotating the axes through an angle of 45 ° .

Solution Summary: The author explains that the graph of the equation sqrtx+.=1 is part of a parabola by rotating the axes through an angle of

To show: The graph of the equation x+y=1 is part of a parabola by rotating the axes through an angle of 45° .

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Chapter 11 Solutions