Question

Please help...
For complex numbers z satisfying
arg( z − u) = π/4,
find the least possible value of |z|.
u = -3i

Please show me how to solve this GEOMETRICALLY (with a half-line).

Answer 1

z- u= a+bi - (-3i)= a+(b+3) i
and tan (b+3)/a = 1

I would start with that, and start drawing figures

Answer 2

I marked the point u at (0,-3) then drew a line from that point at a 45 deg angle to the horizontal axis. Now what do I do to find |z|?

Answer 3

\[ z-u = A e^{i \frac{\pi}{4}} \]
if we plot that , we get the ray