Solve Absolute Values: z∈C |z - 1| = 5, |z - 4| = 4, |4 - z2| = z

[Hannisch]

Homework Statement

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

Homework Equations

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 realized that the absolute value of something 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.

 

[ehild]

i] is correct.

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

ehild

 

[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?

 

[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!

 

[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

 

[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.