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(a) Let the function [tex]$f$[/tex] be defined on the complex numbers as [tex]$\[f(z) = (1+i)z.\]$[/tex] Prove that the distance between [tex]$f(z)$[/tex] and [tex]$0$[/tex] is a constant multiple of the distance between [tex]$f(z)$[/tex] and [tex]$z$[/tex], and find the value of this constant.


(b) Let the function [tex]$g$[/tex] be defined on the complex numbers as [tex]$\[g(z) = (a + 2 i)z\]$[/tex] for some real value of [tex]$a$[/tex]. Then if [tex]$g(z)$[/tex] is equidistant from [tex]$0$[/tex] and [tex]$z$[/tex] for all [tex]$z$[/tex], what is [tex]$a$[/tex] equal to?

Respuesta :

sqdancefan

Answer:

  (a) √2

  (b) 1/2

Step-by-step explanation:

(a) The ratio of distances will be the magnitude of the ratio f(z)/(f(z) -z), so will be ...

  |(1+i)z/((1+i)z -z)| = |(1+i)/i| = |1-i| = √(1+1) = √2

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(b) You want |g(z) -0| = |g(z) -z|, so you have ...

  |(a +2i)z -0| = |(a +2i)z -z|

  |(a +2i)|·|z| = |a-1 +2i|·|z|

Dividing by the magnitude of z and squaring both sides, we have ...

  a^2 +4 = (a-1)^2 +4

  0 = -2a +1 . . . . . . subtract a^2+4 and simplify

  1/2 = a . . . . . . . . . divide by -2, add 1/2