Question

Show that the set of complex numbers z which satisfy |z| = |z - 1 +2i|, (where i = sqrt(-1)), is a line in the argand plane and sketch this line.

Please show steps.
Many thanks

Answer 1

let \(z=a+bi\) then
\[ \large |z|=|z-1+2i| \]
\[ \large |a+bi|=|a+bi-1+2i|=|(a-1)+(b+2)i| \]
\[ \large \sqrt{a ^2+b^2}=\sqrt{(a-1)^2+(b+2)^2} \]

Answer 2

do u get this?

Answer 3

One second please

Answer 4

Do you mind explainig please

Answer 5

since u dont know z u make it ANY complex number
that is the first sentence

Answer 6

ok

Answer 7

now u substitute this into the equation and apply the definition of modulus of a complex number

Answer 8

I understand, yes

Answer 9

is there more to follow?

Answer 10

can u go on by yourself?

Answer 11

I can do the arithmetic, no problem but I have two questions
1)do i replace a and b with integers.
2) And do sketch a normal argand diagram?

Answer 12

1) NO. a and b are just variables. u could have just as well used x and y. plus u are asked for an equation.
2) the argand plane is just the cartesian plane. remember \(\mathbb{C}=\mathbb{R}^2\)

Answer 13

tell me whay u get

Answer 14

one minute please

Answer 15

I am a bit stuck\[\sqrt{a ^{2}+b ^{2}}=\sqrt{a ^{2}-2a+b{^2}+4b+4}\] is what I get

Answer 16

OK let's do it
first u cancel both square roots
\[ \large a^2+b^2=(a-1)^2+(b+2)^2 \]
now expand
\[ \large a^2+b^2=a^2-2a+1+b^2+4b+4 \]
now cancel like items
\[ \large 0=-2a+1+4b+4=-2a+4b+5 \]

Answer 17

understand?

Answer 18

okay

Answer 19

this is your equation

Answer 20

that is the 1st part
many thanks

Answer 21

but how would you sketch that on the argand plane, since it does not have an imaginary part

Answer 22

remember \(z=a+bi\)

Answer 23

a is the real part and b is the imaginary part

Answer 24

okay and the 5?

Answer 25

it is just a constant
u can rewrite the equation into the form
\[ \large b=ma+c \]

Answer 26

like in analytic geometry

Answer 27

okay , nice one, helder_edwin, under which specific section would I find this topic should I wish to read further? You have been great - an officer and a gentlemen!

Answer 28

every book on complex analysis has a first chapter about complex numbers

Answer 29

thank you for your words

Answer 30

okay, I have a 1st year (university) engineering maths book, I will go through it again to search for this particular problem.
I will write you a good testimonial. Thank you so much once and goodbye.

Answer 31

this might help

http://uploading.com/files/5abe2cc8/3540939822Complex%2BAnalysisB.rar/

Answer 32

DL is a bit tricky , its asking me to dl an app? Is that correct?

Answer 33

on the left of the screen there is a counter stopped at 60
and a button that says start timer
press it
it will start counting down when it reaches zero a window will pop up to ask u what to do with the file

Answer 34

or if u want, i can mail the book to u

Answer 35

Thanks for the offer - Its busy DLing right now. It was more tricky than usual :)
How long have you been on open study helder?

Answer 36

i dont know
less than a month
how about u?

Answer 37

I registered some months back but just started using it a lot about two weeks ago. Its really great and what fascinates me is that the help is very fast and free! Unbelievable !

Answer 38

yeah it a great place

Answer 39

Well helder, I am knackered and want to retire to bed. Its 3am here in Galway, West of Ireland, so I need to get some shut eye. Are you Stateside?

Answer 40

i am in Bolivia, south america

Answer 41

rest well
it was nice meeting you

Answer 42

Wow, lovely. Well I hope to catch you again sometime here on openstudy - compadre. Keep well. Adios Amigo. :)

Answer 43

adios