Question

what is the modulus of 1/1+3i - 1/1-3i​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

If z = a + ib is a complex number then modulus of z is given by

$\sf{ \: |z| = |a + ib| = \sqrt{ {a}^{2} + {b}^{2} } \: }$

TO DETERMINE

The modulus of the complex number

$\displaystyle \sf{ \frac{1}{1 + 3i} - \frac{1}{1 - 3i} \: }$

CALCULATION

$\displaystyle \sf{ \frac{1}{1 + 3i} - \frac{1}{1 - 3i} \: }$

$= \displaystyle \sf{ \frac{(1 - 3i) - (1 + 3i)}{(1 + 3i \: )(1 - 3i)} \: }$

$= \displaystyle \sf{ \frac{1 - 3i - 1 - 3i}{ {(1)}^{2} - {(3i)}^{2} } \: }$

$= \displaystyle \sf{ \frac{- 6i }{ 1 - 9 {i}^{2} } \: }$

$= \displaystyle \sf{ \frac{- 6i }{ 1 + 9} \: }$

$= \displaystyle \sf{ - \frac{6i }{ 10} \: }$

$= \displaystyle \sf{ - \frac{3i }{5} \: }$

So

$\bigg | \: \displaystyle \sf{ \frac{1}{1 + 3i} - \frac{1}{1 - 3i} \: } \: \bigg |$

$= \displaystyle \sf{ \bigg | - \frac{3i }{5} \: \: \bigg| }$

$= \displaystyle \sf{ \frac{3 }{5} \: }$