ANOTHERr complex number question. Need help please. (1 Viewer)

ofwgkta · Nov 5, 2018

Hey Been struggling with 2 questions from Patel.

"For the following, describe the locus of the complex number w, where z is restricted as indicated.
Question 1
w=(z-2i)/(1-z), |z|=2
Question 2
w=(z-2+i)/(z+2-i) , |z|=1

Thanks

InteGrand · Nov 5, 2018

ofwgkta said:
Hey Been struggling with 2 questions from Patel.

"For the following, describe the locus of the complex number w, where z is restricted as indicated.
Question 1
w=(z-2i)/(1-z), |z|=2
Question 2
w=(z-2+i)/(z+2-i) , |z|=1

Thanks

$\noindent Here's a way to do the first one. Hopefully you can then try the second one.$

$\noindent We have $w = \frac{z-2i}{1-z}$, so $w-wz = z - 2i \Rightarrow z= \frac{w+2i}{1+w}$. Therefore,$

$\begin{align*}|z|&= 2 \\ \Rightarrow \left|\frac{w+2i}{1+w} \right| &= 2 \\ \Rightarrow |w + 2i|^{2} &= 4 |w +1|^{2}.\end{align*}$

$\noindent Now, write $w=u+iv$ where $u=\mathrm{Re}(w)$ and $v=\mathrm{Im}(w)$, so the equation is$

$\begin{align*} |(u+iv) + 2i|^{2} &= 4 |(u+iv) + 1|^{2} \\ \Rightarrow u^{2} + (v+2)^{2} &= 4\left((u+1)^{2} + v^{2}\right). \end{align*}$

$\noindent From this, you should be able to rearrange the equation into the equation of a certain circle in the $u$-$v$ plane.$

ofwgkta · Nov 5, 2018

InteGrand said:
$\noindent Here's a way to do the first one. Hopefully you can then try the second one.$

$\noindent We have $w = \frac{z-2i}{1-z}$, so $w-wz = z - 2i \Rightarrow z= \frac{w+2i}{1+w}$. Therefore,$

$\begin{align*}|z|&= 2 \\ \Rightarrow \left|\frac{w+2i}{1+w} \right| &= 2 \\ \Rightarrow |w + 2i|^{2} &= 4 |w +1|^{2}.\end{align*}$

$\noindent Now, write $w=u+iv$ where $u=\mathrm{Re}(w)$ and $v=\mathrm{Im}(w)$, so the equation is$

$\begin{align*} |(u+iv) + 2i|^{2} &= 4 |(u+iv) + 1|^{2} \\ \Rightarrow u^{2} + (v+2)^{2} &= 4\left((u+1)^{2} + v^{2}\right). \end{align*}$

$\noindent From this, you should be able to rearrange the equation into the equation of a certain circle in the $u$-$v$ plane.$

Thanks

ANOTHERr complex number question. Need help please. (1 Viewer)

ofwgkta

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InteGrand

Well-Known Member

ofwgkta

New Member

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