Name: Modulus of a Complex Number
Uploaded: 2019-07-19
Duration: 7 min 48 s
Description: Have you heard about the modulus of complex numbers? In this video, let us learn one of the very important parameters of a complex number called the modulus...

Question

Prove that

|\frac{z_{1} - z_{2}}{1 - z_{1} \bar{z_{2}}}| < 1

if

|z_{1}| < 1, and |z_{2}| > 1

or

|z_{1}| > 1

and

|z_{2}| < 1

Accepted Answer

Given, $|\frac{z_{1} - z_{2}}{1 - z_{1} \bar{z_{2}}}| < 1$

$\Rightarrow |z_{1} - z_{2}| < |1 - z_{1} \bar{z_{2}}|$

On squaring both sides of the inequality, we have

${|z_{1} - z_{2}|}^{2} < {|1 - z_{1} \bar{z_{2}}|}^{2}$

We know that $z \bar{z} = {|z|}^{2}$

$\Rightarrow (z_{1} - z_{2}) \bar{(z_{1} - z_{2})} < (1 - z_{1} \bar{z_{2}}) \bar{(1 - z_{1} \bar{z_{2}})}$

$\Rightarrow (z_{1} - z_{2}) (\bar{z_{1}} - \bar{z_{2}}) < (1 - z_{1} \bar{z_{2}}) (1 - \bar{z_{1}} \bar{\bar{z_{2}}})$

$\Rightarrow (z_{1} - z_{2}) (\bar{z_{1}} - \bar{z_{2}}) < (1 - z_{1} \bar{z_{2}}) (1 - \bar{z_{1}} z_{2}) [∵ \bar{\bar{z}} = z]$

$\Rightarrow z_{1} \bar{z_{1}} - z_{1} \bar{z_{2}} - z_{2} \bar{z_{1}} + z_{2} \bar{z_{2}} < 1 - \bar{z_{1}} z_{2} - z_{1} \bar{z_{2}} + z_{1} \bar{z_{1}} z_{2} \bar{z_{2}}$

$\Rightarrow {|z_{1}|}^{2} - z_{1} \bar{z_{2}} - z_{2} \bar{z_{1}} + {|z_{2}|}^{2} < 1 - \bar{z_{1}} z_{2} - z_{1} \bar{z_{2}} + {|z_{1}|}^{2} {|z_{2}|}^{2}$ $[∵ z \bar{z} = {|z|}^{2}]$

$\Rightarrow {|z_{1}|}^{2} + {|z_{2}|}^{2} < 1 + {|z_{1}|}^{2} {|z_{2}|}^{2}$

$\Rightarrow {|z_{1}|}^{2} - {|z_{1}|}^{2} {|z_{2}|}^{2} + {|z_{2}|}^{2} - 1 < 0$

$\Rightarrow {|z_{1}|}^{2} (1 - {|z_{2}|}^{2}) - 1 (1 - {|z_{2}|}^{2}) < 0$

$\Rightarrow ({|z_{1}|}^{2} - 1) (1 - {|z_{2}|}^{2}) < 0$

$\Rightarrow ({|z_{1}|}^{2} - 1) < 0 and (1 - {|z_{2}|}^{2}) < 0$ or $({|z_{1}|}^{2} - 1) > 0 and (1 - {|z_{2}|}^{2}) > 0$

$\Rightarrow {|z_{1}|}^{2} < 1 and {|z_{2}|}^{2} > 1$ or $\Rightarrow {|z_{1}|}^{2} > 1 and {|z_{2}|}^{2} < 1$

$\Rightarrow |z_{1}| < 1$ and $|z_{2}| > 1$ or $\Rightarrow |z_{1}| > 1 and |z_{2}| < 1$

Prove that $|\frac{z_{1} - z_{2}}{1 - z_{1} \bar{z_{2}}}| < 1$ if $|z_{1}| < 1, and |z_{2}| > 1$ or $|z_{1}| > 1$ and $|z_{2}| < 1$

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Important Questions on Complex Numbers and Quadratic Equations

Learn and Prepare for any exam you want !

Prove that $|\frac{z_{1} - z_{2}}{1 - z_{1} \bar{z_{2}}}| < 1$ if $|z_{1}| < 1, and |z_{2}| > 1$ or $|z_{1}| > 1$ and $|z_{2}| < 1$