# Complex Number (Locus)

The question is

A complex number $z$ satisfies

$$| z - 2 + i | = 3$$

(i) Sketch the locus of points that represent $z$ on the Argand diagram.

For this part I drew a circle with the equation

$$(x-2)^2 + (y+1)^2 = 9$$

Therefore a circle with radius $3$ and centre $( 2 , -1 )$.

This is the part I'm stuck on

(ii) What is the maximum value of $\operatorname{Re}(z)$?

The answer states the maximum value is $2$

However when looking at desmos it states the maximum value is $4.828$? • The maximum value of $\text{Re}(z)$ is definitely $5$. Consider $z = 5-i$. Perhaps the problem meant to ask for the maximum value of $\text{Im}(z)$, which is $2$. – JimmyK4542 Dec 20 '15 at 21:02
• The maximum of $\Re(z)$ is $5$. The point $(5,-1)$ is on the circle. $2$ is the maximum of $\Im (z)$. – GaussTheBauss Dec 20 '15 at 21:03

## 1 Answer

Your second problem:

$$|z-2+i|=3\Longleftrightarrow$$ $$z-2+i=\pm3\Longleftrightarrow$$ $$z=\pm3+2-i\Longleftrightarrow$$ $$z=\begin{cases}3+2-i\\ -3+2-i\end{cases}\Longleftrightarrow$$ $$z=\begin{cases}5-i\\ -1-i\end{cases}$$

In general:

$$|z-2+i|=3\Longleftrightarrow$$ $$z=(2-i)+3e^{ni}\space\space\space\space\text{with}\space n\in\mathbb{R}$$

Because, if $n\in\mathbb{R}$:

$$|((2-i)+3e^{ni})-2+i|=|3e^{ni}|=3$$

• What do the arrows represent? (⟺) – bigfocalchord Dec 20 '15 at 21:14
• @dydxx Solving an equation in steps. – Jan Dec 20 '15 at 21:16
• Ah okay , sorry if this a stupid question but I don't really understand what z = 5-i , z = -1 - i represent? @JanEerland – bigfocalchord Dec 20 '15 at 21:16
• @dydxx You're totally welcome! – Jan Dec 20 '15 at 21:17