If |z|=2 then -1+5z lie on1).circle2).parabola .3).hyperbola

z’ = 1+5z and so z’+1 = 5z which means |z’+1| = 5|z| = 10 which means z’ lies on a circle centred at (-1,0) and radius 10.

 

Alternatively you can see that 5z takes all points on the circle |z| =2 to the circle |z|=10 (it expands the circle). Then 5z -1 is simply displacing the circle left by 1 unit...so the figure remains a circle. This has the advantage of noticing that the whole process is reversible – you can map points on both circles bijectively to each other