FP3 Complex numbers

Hi,

I'm told:

$w = \dfrac{2}{z - 2i}$

I'm asked to show that:

$| z | = 2 \begin{vmatrix} \dfrac{w-i}{w} \end{vmatrix}.$

I did this by saying:

$w = \dfrac{2}{z - 2i}$

$\Rightarrow z - 2i = \dfrac{2}{w}$

$z = \dfrac{2+2wi}{w}$

$\Rightarrow z = 2(\dfrac{1+wi}{w})$

This is where I'm not too sure I've shown what I'm asked to sufficiently. I say this:

$z = 2\begin{bmatrix} \dfrac{-\frac{1}{i}(w-i)}{w} \end{bmatrix}$

As $\begin{vmatrix} -\dfrac{1}{i} \end{vmatrix} = 1$

$\Rightarrow \begin{vmatrix}z \end{vmatrix} = 2\begin{vmatrix} \dfrac{w-i}{w} \end{vmatrix}.$

Would this be a sufficient "show" answer, or does it seem like I've winged it?

Thank you

Hi,


Would this be a sufficient "show" answer, or does it seem like I've winged it?

I personally think what you've done is acceptable

It's a perfect answer.