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 $4.828$?
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 $n\in\mathbb{R}$:
$$|((2-i)+3e^{ni})-2+i|=|3e^{ni}|=3$$