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

#### ofwgkta

##### New Member
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

##### Well-Known Member
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
$\bg_white \noindent Here's a way to do the first one. Hopefully you can then try the second one.$

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

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

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

\bg_white \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*}

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

#### ofwgkta

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

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

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

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

\bg_white \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*}

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