help plz (2 Viewers)

[mathsbrain]

How do you graph |z-1/z|=2?

 

[Eagle Mum]

I’m not a maths expert, but nobody else has responded, so this is how I would approach this question:

As a maths extn 2 question, I gather that ‘z’ is a complex number of the form x + iy

|z - 1/z| = 2

z - 1/z = 2 or z - 1/z = -2

which produces two quadratic equations:

z^2 - 2z - 1 = 0 and z^2 + 2z - 1 = 0

and four possible values of z and therefore x (since z has no imaginary component in any of these solutions):

1 + sqrt (2) and -1 + sqrt (2)

Hence, I would plot these values on the x - axis with y = 0 for each point.

I don’t know how I would prove there are no solutions with non zero values of y, nor how I would find such solutions if they exist.

 

[C2H6O]

This is a really tough question and I'm sure there's some niche way to solve it, but brute forcing it by subbing z=x+iy is always an option








Now I found this factorisation on the internet, but you definitely wouldn't be able to find this normally

By null factor theorem, you get 2 equations:


The graph is 2 circles, centred at (0, 1) and (0, -1), both with radius root 2