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

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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)]1/2 = 3

 

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

 

(x-5)2 + (y-3) = 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

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