complex numbers help (1 Viewer)

[m2]

does anyone know how 2 do locus questions for comples numbers like this one

arg(z - x) - arg(z) where x is any number? if u can, could u xplain how 2 do it.

thanx.

 

[.ben]

oh yeh. isn't arg (z-x) just the same thing with shifted to the right x units?

 

[Stefano]

I think:

z-x = x+iy-x = iy

So, Arg(z-x) = pi/2 (since its purely imaginary)

So, Arg(z-x) - Arg(z) = (pi/2) - [tan(y/x)] ??

That right? Sorry, I'm in a hurry.

 

[Slidey]

Stefano: x is actually a complex number of the form u+vi

for z=a+bi, z-x=a-u+(b-v)i
arg(z-x)-arg(z)=arctan([b-v]/[a-u])-arctan(b/a)=c
Take the tan of all that:
tan{arctan([b-v]/[a-u])-arctan(b/a)}
=([b-v]/[a-u]-b/a)/(a^2-au+b^2-bv)/[a^2-au])
=(bu-av)(a^2-au)/(a^2-au+b^2-bv)=tan(c)

Meh. Anyway, I think the answer is the major segment of a circle. I remember Will Hunting went nuts over this question once.