Question

If, with an Argand diagram with origin O, the point Prepresents Z and Q represents
1/Z* (Z*=Z conjugate), prove that O,P and Q are collinear and find the ratio OP:OQ in terms of |Z|

Answer 1

OK @powerangers69 I want you to draw a diagram first. Go ahead!

Answer 2

The complex number Z = a + bi where a is the real part and b is the imaginary part. The conjugate of a complex number is determined by multiplying the imaginary part by negative one. Thus the conjugate of Z = a + bi is Z* = a - bi. Conversely the conjugate of Z* = a - bi is Z = a + bi. In other words, the conjugate of the conjugate of the complex number is the given complex number.

On the complex plane, the real axis is the horizontal axis, and the imaginary axis is the vertical axis.

A complex number is represented on the complex plane by plotting the point (a, b). Consequently, the conjugate of the complex number on the complex plane is represented by plotting (a, -b).

If point P represents the complex number Z, then let P(a, b). If Q represents the reciprocal of the conjugate of Z, then Q(a/(a² + b²), b/(a² + b²)). You ask how I found Q? Please allow me to explain.

Q is the reciprocal of the conjugate of Z = a + bi, correct? The conjugate of Z is

Z* = a - bi

The reciprocal of the conjugate of Z is 1/Z* or 1/(a - bi). This reciprocal is not in proper form. Thus we rationalize the denominator. Allow me to demonstrate.\[\frac{ 1 }{ a -bi }⋅\frac{ a +bi }{ a +bi }=\frac{ a +bi }{ a ^{2}+b ^{2} }=\frac{ a }{ a ^{2}+b ^{2}}+\frac{ b }{ a ^{2}+b ^{2} }i =\frac{ 1 }{ a ^{2}+b ^{2} }(a +bi)\]Thus Q(a/(a² + b²), b/(a² + b²)). Since O is the origin, vector OP = (a, b) and
vector OQ = (a/(a² + b²), b/(a² + b²)).

Two vectors are collinear iff one is a scalar multiple of the other. For example, vector u and vector v are collinear vectors iff u = kv where k is some scalar (any real number).

\[OQ =\frac{ 1 }{ a ^{2}+b ^{2} }OP\]∴ OP and OQ are collinear.