Htykuut

I've to sketch the complex number $z$ such that it satisfy both the inequality
$|(z-2i)|le2$ and $ 0le arg(z+2)le 45^circ$

I was able to sketch and shade the region that satisfies both inequality, Please see the below link.

Sketch on Argand diagram

I've a problem in getting the greatest value of $|z|$ (maximum length of $z$ from $(0,0),$ it should come around 3.70)

if you parametrize the circle of constraint it is $(2cos t,2sin t +2)$ with $t=0$ corresponding to the point $(2,2)$. so:

$$
frac18 |z|^2 = frac12 (cos^2 t +sin^2 t + 2 sin t +1 )= 1+ sin t
$$

Inspection of the figure indicates that the point $z_0:=sqrt{2}+i(2+sqrt{2})$ is the feasible point with largest absolute value: The closed disk $D_0$ with center $0$ and radius $|z_0|$ obviously contains the complete feasible region $F$, and $z_0in F$. It follows that
$$max_{zin F}|z|=|z_0|=sqrt{8+4sqrt{2}}doteq3.6955 .$$

Let $0+2i$ be C (the centre of the circle), let the point of furthest intersection (greatest $|z|$) of the 2 loci be A (You have indicated it on your diagram with a cross), construct the point B such that $triangle CAB$ has a right angle at B.

From here we can note that $angle CAB=frac{pi}{4}$(given - construction of CB is such that it is parallel to the x-axis, use corresponding angles), $AC=2$ (radii). Using trigonometry we can find that $BC=2cos(frac{pi}{4})=sqrt{2}$, similarly $AB=2sin(frac{pi}{4})=sqrt{2}$.

Applying our knowledge to the diagram (the Argand diagram), we see that the real part of the desired complex number is given by $BC$, $Re(z)=sqrt{2}$. The imaginary part is the sum of $AB$ and the height (distance) from the number $sqrt{2}$ to $sqrt{2}+2i$ (distance is $2$), thus $Im(z)=BC+2=2+sqrt{2}$.

$therefore$ $z=sqrt{2}+i(2+sqrt{2})$ and $|z|=2sqrt{2+sqrt{2}}$