Explanation Given: The given equation is, x + y = 1 Rotate the axes by an angle of 45 ∘ . Approach: The standard equation of a conic is, A x 2 + B x y + C y 2 + D x + E y + F = 0 . Suppose the x − and y − axes in a coordinate plane are rotated through the acute angle ϕ to produce the X − and Y − axes. Then, the coordinates ( x , y ) and ( X , Y ) of a point in the X Y − and x y − planes are related as, x = X cos ϕ − Y sin ϕ … ( 1 ) y = X sin ϕ + Y cos ϕ … ( 2 ) X = ( x cos ϕ + y sin ϕ ) Y = ( − x sin ϕ + y cos ϕ ) Calculation: First, remove the radicals from the given equations, ( x + y ) 2 = 1 2 x + y + 2 x y = 1 ( 2 x y ) 2 = ( 1 − ( x + y ) ) 2 4 x y = 1 + ( x + y ) 2 − 2 ( x + y ) 1 + x 2 + y 2 + 2 x y − 2 x − 2 y = 4 x y 4 x y − 1 − x 2 − y 2 − 2 x y + 2 x + 2 y = 0 x 2 − 2 x y + y 2 − 2 x − 2 y + 1 = 0 ( 3 ) Equation ( 3 ) represents the given equation after radicals are removed. The axes have to be rotated by an angle of 45 ∘ . Substitute 45 ∘ for ϕ in equations ( 1 ) and ( 2 ) to get, x = X cos ( 45 ∘ ) − Y sin ( 45 ∘ ) = X ( 1 2 ) − Y ( 1 2 ) = X − Y 2 … ( 4 )

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN: 9781305071742

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

See similar textbooks

Question

Chapter 12.5, Problem 36E

To determine

To show:

The equation represents a parabola by rotating the axes.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.5, Problem 35E

Chapter 12.5, Problem 37E

Students have asked these similar questions

How long is a guy wire reaching from the top of a 15-foot pole to a point on the ground 9-feet from the pole? Question content area bottom Part 1 The guy wire is exactly feet long. (Type an exact answer, using radicals as needed.) Part 2 The guy wire is approximatelyfeet long. (Round to the nearest thousandth.)

Question 6 Not yet answered Marked out of 5.00 Flag question = If (4,6,-11) and (-12,-16,4), = Compute the cross product vx w k

Consider the following vector field v^-> (x,y): v^->(x,y)=2yi−xj What is the magnitude of the vector v⃗ located in point (13,9)? [Provide your answer as an integer number (no fraction). For a decimal number, round your answer to 2 decimal places]

Chapter 12 Solutions

Algebra and Trigonometry (MindTap Course List)

Ch. 12.1 - A parabola is the set of all points in the plane...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E Ch. 12.1 - Prob. 5E Ch. 12.1 - Prob. 6E Ch. 12.1 - 5-10 Graphs of Parabolas Match the equation with...Ch. 12.1 - Prob. 8E Ch. 12.1 - 5-10 Graphs of Parabolas Match the equation with...Ch. 12.1 - Prob. 10E

Knowledge Booster

The equation represents a parabola by rotating the axes.

Solution Summary: The author explains how the given equation represents a parabola by rotating the axes.

To show: The equation represents a parabola by rotating the axes.

Want to see the full answer?

Chapter 12 Solutions

To show:

The equation represents a parabola by rotating the axes.