Understanding Complex Plane Regions: Solving Equations and Graphing Circles

[goojilla]

Homework Statement

Shade each region in the complex plane. Justify your solution.

1) z - Conjugate[z] = 4

2) 1 + z, where |z| = 1

The Attempt at a Solution

So for my attempt for 1 is:

Let z = x + iy therefore Conjugate[z] = x - iy

z - Conjugate[z] = 4
x + iy - (x - iy) = 4
2iy = 4
iy = 2 ***multiply both sides by i
-y = 2i
y = -2i

Now from my understanding y is supposed to be a real number, is it not? So what exactly does
y = -2i represent? What region would it be in the complex plane?

And my attempt for 2:

|z| = 1 represents a circle of unit radius 1.
Given 1+z and |z|=1, I changed this to into the equation of a disk in the complex plane (can I even do that?)

Let z = x + iy
|z+1| < 1 ***For this step would I use less than, less than or equal to, or just equal to?
|x+iy+1| < 1
(x+1)² + y² < 1²

Now this gives me an equation for a circle of radius 1 centred around the point (-1,0)
Is this correct?

Thanks

 

[Dick]

Homework Statement

Shade each region in the complex plane. Justify your solution.

1) z - Conjugate[z] = 4

2) 1 + z, where |z| = 1

The Attempt at a Solution

So for my attempt for 1 is:

Let z = x + iy therefore Conjugate[z] = x - iy

z - Conjugate[z] = 4
x + iy - (x - iy) = 4
2iy = 4
iy = 2 ***multiply both sides by i
-y = 2i
y = -2i

Now from my understanding y is supposed to be a real number, is it not? So what exactly does
y = -2i represent? What region would it be in the complex plane?

And my attempt for 2:

|z| = 1 represents a circle of unit radius 1.
Given 1+z and |z|=1, I changed this to into the equation of a disk in the complex plane (can I even do that?)

Let z = x + iy
|z+1| < 1 ***For this step would I use less than, less than or equal to, or just equal to?
|x+iy+1| < 1
(x+1)² + y² < 1²

Now this gives me an equation for a circle of radius 1 centred around the point (-1,0)
Is this correct?

Thanks

z-conjugate(z)=4 has no solutions in the complex plane, as you've correctly deduced. So there is no region. It's just the empty set. For the second part, if |z|=1 is a circle around z=0 of radius 1 then isn't |z-c|=1 a circle around z=c of radius 1? |(z+1)-1|=|z|=1.

 

[goojilla]

I do not understand how z - conjugate(z)=4 has no solutions in the complex plane. Can you explain that?

 

[Dick]

I do not understand how z - conjugate(z)=4 has no solutions in the complex plane. Can you explain that?

You explained it. z-conjugate(z) is a purely imaginary number, it's 2yi where y is real. 4 is real and not zero. They can't be equal.

 

[goojilla]

Okay that makes sense, thank you!