Question

The complex numbers z and w are represented, respectively, by the points P(x,y) and Q(u,v) respectively in Argand diagrams and w=1/z
(a) show that x=u/(u^2+v^2)
and find an expression for y in terms of u and v.

Answer 1

I would use the following,
\[w=1/z\Rightarrow z=1/w\\ z=\frac{\overline{w}}{|w|^2}=\frac{u-iv}{u^2+v^2}\]
\[x+iy=\frac{\overline{w}}{|w|^2}=\frac{u}{u^2+v^2}-\frac{iv}{u^2+v^2}\Rightarrow\\
\Rightarrow x=\frac{u}{u^2+v^2}\]

Answer 2

zw = 1

z= x+iy
w= u +vi

(u+vi)(x+iy) = 1

ux +iyu +ivx -vy = 1
ux - vy =1 -----1
uy + ux = 0-----2

make y the subject in eq 2 = y = -xv/u---4

sub (y) in 4 to equation 1 and solve u get it

Answer 3

for the second part make x the subject in equation 2
and substite x in equation 1 u should get

\[y = \frac{-v}{v^{2}+u^{2}}\]

Answer 4

Oh, got it. Thank you. @John_ES @thushananth01 ^_^

Answer 5

Who do I give the medal to now? >_<

Answer 6

@emcrazy14 how do you do this http://screencast.com/t/YCDyJ0fU

Answer 7

What did you get for w^2?

Answer 8

3 +4i

Answer 9

This will be a circle with centre at (3,4) and with radius 5. You'll shade the inner region of the circle.

Answer 10

what about the other inequality