Help with complex loci on Argand diagram

[stfz]

Hi all, I'm hoping for some help/guidance on this problem:
1. Homework Statement
The complex number z is represented by the point P on the Argand diagram. Sketch and describe in words the locus of P if:

i) ##|2-iz|=2##
ii) ##|\dfrac{1+i}{z}-2|=2##

The solutions read:
i) ##|2-iz|=2 \Leftrightarrow |z+2i|=2##
ii) ##|\frac{1+i}{z} -2| = 2 \Leftrightarrow |z - \frac{1+i}{2}| = |z|##

In (i), apparently what they've done is multiplied the expression ##2-iz## within the modulus by ##i##. But it doesn't explain why that is valid...
With (ii), I have no idea how they did that...
Some help here would be appreciated - I haven't seen any formal information given on this in the book, and I can't see how the result is implied :(

Homework Equations

##|z-a|## represents the scalar/real distance between two complex numbers ##z## and ##a## on the Argand diagram.

The Attempt at a Solution

(i) - Multiplication of expression inside modulus by ##i## yields answer, but I don't have the proof for why this can be done.
(ii) - Not really sure how to start here.

Thanks!
Stephen :)

 

[NascentOxygen]

Multiplying a complex number by i amounts to a rotation of +90 degrees on the Argand plane, but doesn't change the vector's length (modulus). Plot 5+i2 and -2+i5 to illustrate this.

So perhaps in the interest of simplifying the expression, multiplying by i inside a modulus is a useful technique?

 

[Ray Vickson]

Hi all, I'm hoping for some help/guidance on this problem:
1. Homework Statement
The complex number z is represented by the point P on the Argand diagram. Sketch and describe in words the locus of P if:

i) ##|2-iz|=2##
ii) ##|\dfrac{1+i}{z}-2|=2##

The solutions read:
i) ##|2-iz|=2 \Leftrightarrow |z+2i|=2##
ii) ##|\frac{1+i}{z} -2| = 2 \Leftrightarrow |z - \frac{1+i}{2}| = |z|##

In (i), apparently what they've done is multiplied the expression ##2-iz## within the modulus by ##i##. But it doesn't explain why that is valid...
With (ii), I have no idea how they did that...
Some help here would be appreciated - I haven't seen any formal information given on this in the book, and I can't see how the result is implied :(

Homework Equations

##|z-a|## represents the scalar/real distance between two complex numbers ##z## and ##a## on the Argand diagram.

The Attempt at a Solution

(i) - Multiplication of expression inside modulus by ##i## yields answer, but I don't have the proof for why this can be done.
(ii) - Not really sure how to start here.

Thanks!
Stephen :)

In (i) they used two facts:
(1) ##|z_1 z_2| = |z_1| \; |z_2|## for any two complex numbers ##z_1, z_2## (applied to ##z_1 = i##, ##z_2 = 2-iz##).
(2) ## i \times -i = +1##.

 

[stfz]

Thanks for the help :) I have figured it out. Thanks again :D Just proved it myself that if ##|z_1| = |z_2|##, then ##|kz_1|=|kz_2|, k\in C \cup R##