# Complex numbers VI

#### Punch

##### New member
Sketch on an Argand diagram the set of points satisfying both |z-4i|<=\sqrt{5} and \frac{\pi}{4}<=arg(z+4)<=\frac{\pi}{2}.

I have already sketched the 2 loci. The problem lies in the following part.

Hence find the least value of |z-2\sqrt{2}-4i|. Find, in exact form, the complex number z_1 represented by the point P that gives this least value.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Hence find the least value of |z-2\sqrt{2}-4i|.
Over which set of z? The intersection defined in the first part? It seems that you need to find the shortest distance from a point to a semi-disk.

Why don't you wrap the $$...$$ tags around your formulas?

#### Punch

##### New member
Over which set of z? The intersection defined in the first part? It seems that you need to find the shortest distance from a point to a semi-disk.

Why don't you wrap the $$z_1$$ tags around your formulas?
Yes, how do I then find the complex number $$z_1$$ in the following part?

I tried using the latex but they didnt seem to work

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Yes, how do I then find the complex number $$z_1$$ in the following part?
See the following picture.

I tried using the latex but they didnt seem to work
Type $$\frac{\pi}{4}\le\arg(z+4)\le\frac{\pi}{2}$$ to get $$\frac{\pi}{4}\le\arg(z+4)\le\frac{\pi}{2}$$.