Complex-Theory-Geometric-Interpretations

MATHEMATICIAN

Complex Theory
Geometric interpretations

A Level

STELLA
SEREMETAKI

Stella Seremetaki
http://www.mathschool-online.gr
Complex Theory - Geometric interpretations
1 Exercise
If for the complex z is valid z  3 calculate the
presentation A z 3i 2 z 3i 2
Give a geometric interpretation for the case z  3i
Solution

A  z  3i 2  z  3i 2  z  3i  z  3i   z  3i  z  3i 

I perform the actions in a partition z  3 thus Α=36
(1)
Let N, K the images of z, (z  3i ) , 3i,-3i
respectively.
Μ is a point of the circle (Κ(0,0),ρ=3)

1

Stella Seremetaki
http://www.mathschool-online.gr

I notice that

z  3i = MN, z  3i  MK 
z  3i 2  z  3i 2  MN2  MK 2  36
KN  3i  3i  6i  6  KN2  36
Thus MN2  MK 2  (KN)2

So the geometric interpretation A  z  3i 2  z  3i 2
is the rectangular triangle NMK
Note
The above conclusion is to be expected because the
NMK angle is recorded in the circle (K (0,0), 3) and
goes to a semicircle, so it is correct and the triangle
NMK is right
Excercise
2. Find the geometric location of the complex z
images for which it applies z  3  z  5  3i
Solution
z  3  z  5  3i  z  3  z (5  3i )
Let N, K be the images of the complex ones z , 3 , 5-3i
That is Ν(3,0),Κ(5,-3) .

2

Stella Seremetaki
http://www.mathschool-online.gr
I should find the points N and K at the complex
level

As well as the median of the NK segment
z  3  z  5  3i  z  3  z (5  3i )

The above equation expresses the images of the
complex z that are located at the semi-plane
containing the edge of the median of the NK which
contains the point N (0,3) as well
Hence the geometric locus is the semi-plane with the
acne of the median of NK which contains the point
N(3,0)

Thank you!

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