Question

PLEASE HELP.
Complex numbers. *question attached below* Will give medal and fan :)

Answer 1

I'm pretty much stuck on this problem :( Can I have some help, please?

Answer 2

yes

Answer 3

nope

Answer 4

Notice that reducing the imaginary component by 1 unit is giving the same length as reducing the real component by 1 unit

Answer 5

Can we say the locus of z has to be that straight line through origin ?

Answer 6

How do we know it passes through the origin? :S

Answer 7

plugin z = 0+0i in the given equation

Answer 8

Oooh, but aren't we supposed to plug in z=x+iy instead of 0+0i?

Answer 9

Ahh that works too but I thought this is easy to figure out with geometry argument,
lets solve it algebraically now :)

Answer 10

|z-i| = |z-1|

|x+i(y-1)| = |(x-1) + iy|

x^2 + (y-1)^2 = (x-1)^2 + y^2

simplify

Answer 11

x^2+y^2-2y+1 = x^2-2x+1+y^2

Answer 12

So we end up with 2x-2y=0?

Answer 13

yes cancel 2 and you would get y=x

Answer 14

Yes :)

Answer 15

Do we do the same thing for |w-4|=2 now?

Answer 16

we caan, but again doesn't that equation look closely related to r = 2 ?

Answer 17

it seems the center got shifted right by 4 units, otherwise it should give a circle of radius 2 units eh ?

Answer 18

Oh yes

Answer 19

So on the Argand diagram, do we have a straight line and a circle, then?

Answer 20

yes draw them