Calculate the value of |z|
Level
2
Re(any complex number) refers to the real part of a complex number ie, Re(X + iY) = X.
If Z is a complex number , it is given that Re ((1-Z)/(1+Z)) = 0 , What is the value of |Z|.
|Z| refers to Mod(Z).
The answer is 1.
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1 solution
Let z = x + iy; 1-z/1+z can be written as ( 1-x - iy)/(1+x + iy) ;
Multiplying both numerator and denominator by 1+x-iy we have the expression
((1-x-iy) * (1+ x - iy))/ ((1+x)^2 + y^2)
Let D = ((1+x)^2 + y^2)
= (1-iy + x)(1-iy -x)/D
= ((1-iy)^2 - x^2 )/D
= (1 - y^2 - 2 i y - x^2 )/D
Rearranging the expression as
(1-y^2- x^2)/D -2y*i/D
Since Re((1-z)/(1+z)) = 0 we have
(1-y^2- x^2)/D = 0
or 1-y^2- x^2 = 0
x^2 + y^2 = 1
If z = x + iy where x and y are real and i = √-1. Then the non negative square root of (x^2+ y^2) is called the modulus or absolute value of z (or x + iy)
So Mod(z) or |z| = 1