complex numbers.!!!!!!!!!help plz!!!!!!!!!?

Question

if the complex z=(3*K+i)/(1+K*i) with KεR,

prove this one: 1<=/z/<=3

if /z/ is the modulus.

Answer 1

modulus of z=sqrt(9k^2+1)/sqrt(1+k^2)
to find the maximum and minimum value of the function take y=sqrt(9k^2+1)/sqrt(1+k^2)so we get k^2=(1-y^2)/((y^2-9)
this is =>0 so we write (y-1)(y+1)/(y-3)(y+3) <=0
so the ranges of y are -1=>y=>-3 or 3=>y=>1 but the modulus of a complex number is positive. so the maximum and minimum value of y is 3 and 1

Answer 2

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