How do I sketch the locus of |z - 5-3i | = 3 on an Argand Diagram?

First, we use the idea that a complex number z can be written in terms of its real and imaginary parts, i.e. z = x+iy, to write our expression as:

 

| x+ iy -5 - 3i | = 3

Next, we can group the real and imaginary parts of the above expression, giving us:

| (x-5) + i(y -3) | = 3

 

Now that the expression is in the form a+ib, we can use that the modulus of a complex number is the square root of (a² + b²), to write our expression as:

[ (x-5)² + (y-3)² ]^1/2 = 3

 

Finally, by squaring both sides of the equation, we get:

 

(x-5)² + (y-3)²  = 3²

 

This sort of expression should look familiar to you; it's the standard equation for a circle!  So our final plot on our Argand diagram is of a circle center (5,3) with a radius of 3. By extending the ideas we've considered in this example, it follows that the expression |z- z₁| = r represents a circle centered at z₁ = x₁ + iy₁, with a radius r.