Sketch, on a single Argand diagram, the loci given by $| z − \sqrt 3 − i | = 2$ and $\arg (z) = 16 \pi$

For your first part, let $z=x+iy$ so then we get that $$ | z- (\sqrt{3} + i)| = | (x-\sqrt{3}) + i(y-1)| = (x-\sqrt{3})^2 + (y-1)^2 = 2 $$

which is the locus of a circle with centre $(\sqrt{3},1)$ and radius $ \sqrt{2}$. Substituting $ x= 0, y=0$ gives us that it intersects the origin.

For your second part, consider that arg z = $arctan(\frac{y}{x})$.

$|z-z_0|=r>0$ describes a circle with centre at $z_0$ and radius $r$. $\arg (z-z_0)=\theta$ describes a ray emanating from $z_0$ at an angle of $\theta$ to the positive real axis in the complex plane.

In your example, using Pythagoras theorem, one sees that the origin is indeed at distance $2$ from the point $(\sqrt 3,1)$.

For the first part, take two points in the complex plane (Argand diagram). Their difference is represented in the diagram by the line joining the two points (direction corresponds to sign - ie which we subtract from which).

The length of the line joining the two points is the modulus of the difference.

Your first equation therefore expresses that the distance between the variable point $z$ and the fixed point $\sqrt 3+i$ is equal to 2. The locus of a point at a constant distance from a fixed point is a circle. This circle has radius 2 and centre $(\sqrt 3,1)$ since these are the fixed data you are given.

The circle passes through $(0,0)$ simply because the centre of the circle is distance 2 from the origin (check it) and the radius of the circle is 2.

What does the argument represent in the argand diagram? Can you find any points with $\arg z = 16\pi?$ (or with argument $\pi,2\pi,3\pi \dots?)$