2. Fix xo, Yo, 0 ER and define bijections T, R : R² → R² by the rulesT(r,y)(x – xo, y – Yo)andR(x, y) = (x cos 0 + y sin 6 , x sin 0 – y cos 0) .Thus T is the parallel translation of R? that takes (xo, Yo) to the origin, and Ris the reflection of R? in the line through the origin that makes an angle of 0/2radians with the positive r-axis. (You do not need to verify these facts.)(a) Write down the rules for T-1 and R-1. You will get full credit if you do thiscorrectly, and you do not need to provide any justification.Let C be some general curve in R? defined by the equationf(x, y) = 0,where f(x, y) is some algebraic expression involving x and y, that is,C = {(x, y) E R² | f(x, y) = 0 } .It follows quickly that if B : R² –→ R² is any bijection then the image B(C) ofthe curve C under B is defined by the equationf(B-'(x, y)) = 0.(You do not need to verify this fact.)(b) Deduce from part (a) that T(C), the image of C under T, is the curve definedby the equationf(r + xo, Y+ Yo) = 0and R(C), the image of C under R, by the equationf(x cos 0 + y sin 0, x sin 0 – y cos 0) = 0.%3D(c) Find an equation that defines the curveT-'(R(T(C))) ,that is, the curve that results as the image of C after first applying T, thenapplying R and finally applying T-1.

Answered: Image /qna-images/answer/a324e262-2955-4907-a144-bca1a3445bda.jpg

is the reflection of R² in the line through the origin that makes an angle of 6/2 radians with the positive r-axis. (You do not need to verify these facts.) (a) Write down the rules for T-1 and R-1. You will get full credit if you do this correctly, and you do not need to provide any justification. Let C be some general curve in R² defined by the equation f(r, y) = 0, where f(r, y) is some algebraic expression involving r and y, that is, C = {(x,y) E R² | ƒ(x, y) = 0 } . It follows quickly that if B : R² R² is any bijection then the image B(C) of the curve C under B is defined by the equation f(B-'(r,y)) = 0. %3D (Vou do not nood to rorifr thie foot)

is the reflection of R² in the line through the origin that makes an angle of 6/2 radians with the positive r-axis. (You do not need to verify these facts.) (a) Write down the rules for T-1 and R-1. You will get full credit if you do this correctly, and you do not need to provide any justification. Let C be some general curve in R² defined by the equation f(r, y) = 0, where f(r, y) is some algebraic expression involving r and y, that is, C = {(x,y) E R² | ƒ(x, y) = 0 } . It follows quickly that if B : R² R² is any bijection then the image B(C) of the curve C under B is defined by the equation f(B-'(r,y)) = 0. %3D (Vou do not nood to rorifr thie foot)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Question

2. Fix xo, Yo, 0 ER and define bijections T, R : R² → R² by the rules
T(r,y)
(x – xo, y – Yo)
and
R(x, y) = (x cos 0 + y sin 6 , x sin 0 – y cos 0) .
Thus T is the parallel translation of R? that takes (xo, Yo) to the origin, and R
is the reflection of R? in the line through the origin that makes an angle of 0/2
radians with the positive r-axis. (You do not need to verify these facts.)
(a) Write down the rules for T-1 and R-1. You will get full credit if you do this
correctly, and you do not need to provide any justification.
Let C be some general curve in R? defined by the equation
f(x, y) = 0,
where f(x, y) is some algebraic expression involving x and y, that is,
C = {(x, y) E R² | f(x, y) = 0 } .
It follows quickly that if B : R² –→ R² is any bijection then the image B(C) of
the curve C under B is defined by the equation
f(B-'(x, y)) = 0.
(You do not need to verify this fact.)
(b) Deduce from part (a) that T(C), the image of C under T, is the curve defined
by the equation
f(r + xo, Y+ Yo) = 0
and R(C), the image of C under R, by the equation
f(x cos 0 + y sin 0, x sin 0 – y cos 0) = 0.
%3D
(c) Find an equation that defines the curve
T-'(R(T(C))) ,
that is, the curve that results as the image of C after first applying T, then
applying R and finally applying T-1.

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