Absolute values

1. Sep 20, 2009

Hannisch

1. The problem statement, all variables and given/known data
So I've got two problems I'm struggling a bit with. One of them I've solved (I think), but I'm definitely not sure. The other one is bugging me a bit. Anyway:

i] Determine all z∈C so that |z - 1| = 5 and |z - 4| = 4

ii] Determine all z∈C so that |4 - z2| = z

2. Relevant equations

3. The attempt at a solution
i] I say that z = x+yi as a starting point. From there:

|x + yi -1| = 5
√( (x - 1)2 + y2 ) = 5
x2 + 1 -2x +y2 = 25

|x + yi -4| = 4
√( (x-4)2 + y2 ) = 4
x2 + 16 - 8x + y2 = 16

y2 = 8x - x2

Inserting this in the first equation:

x2 + 1 - 2x + 8x - x2 = 25

6x + 1 = 25

x = 4

and then y2 = 32 - 16 = 16, y = ± 4

So I get z = 4±4i

I think this should be correct, but I'm a bit.. unsure.

ii] I've gotten so far that I've looked at the exercise and realised that the absolute value of someting is always a real number, which means if z = x+yi, then y=0. But from here I'm unsure on how to proceed.

How on earth am I supposed to solve this? I'm feeling.. lost.

2. Sep 20, 2009

ehild

i] is correct.

ii] You are right, z is real. How do you define the absolute value of a real number?

ehild

3. Sep 20, 2009

Hannisch

The only thing I can think of right now (it's.. late) is:

|x| = x if x>0
|x| = 0 if x=0
|x| = -x if x<0

Is this what you mean?

4. Sep 21, 2009

Hannisch

Never mind, I had a insight today during my lecture and suddenly it was all very, very clear and the answers are something like ±(1 + √17)/2

Thanks though!

5. Sep 21, 2009

ehild

Almost good! Do not forget that z can not be negative as it is equal to an absolute value. You had two second order equations, with 4 roots altogether, but only the positive roots are valid. (±1 + √17)/2

ehild

6. Sep 23, 2009

Hannisch

Yeah, sorry, I put the plus/minus sign wrong :) I figured that out and even checked if they were in the right intervals and such.