Toronto Math Forum
MAT3342018F => MAT334Tests => Quiz1 => Topic started by: Victor Ivrii on September 28, 2018, 04:09:13 PM

$\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}$
Describe the locus of points $z$ satisfying the given equation.
\begin{equation*}
 z  i = \Re z.
\end{equation*}

x+yii=Rez
x+(y1)i=x
Square Root [x^2+(y1)^2]=x
x^2 + (y1)^2=x^2
(y1)^2=0
So, y=1

Let $z=x+iy$
Then $Re(z) = x$ and $zi = x+i(y1)$
Thus:
$Re(z) = zi$
$x = x+i(y1)$
$x = \sqrt{x^2 + (y1)^2}$
$x^2 = x^2 +(y1)^2, x \ge 0$
$(y1)^2 = 0, x \ge 0$
$y=1, x \ge 0$
In complex terms:
$ y = 1 \iff Im(z) = 1$ and $ x \ge 0 \iff Re(z) \ge 0$
Thus the equation of the line in complex terms is:
$Im(z) = 1, Re(z) \ge 0$
This is the horizontal half line extending from $z=i$ rightward.

Vedant,
it is a whole line. Also, no need to post after solution is posted