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complex numbers exercise with answer | example 2

 

complex numbers exercise : 

Let the complex plane  in orthogonal coordinates.

For each question, you can make several correct suggestions. Set the correct answers and justify your answer.

1) Let the points A , B , C  Their complex numbers respectively: 

a-2+3i , b-3-i  , c2.08+1.98i .

The triangle ABC is:

  • Isosceles and  not a right triangle.
  • A right triangle and not isosceles.
  • Right triangle and isosceles.
  • A triangle that not right and not isosceles.
2) For each complex number Z ≠ -2 We put the complex number Z' Where:  


Z=(Z4i)Z+2

a) The set of points M with a complex number Z where: |Z'|=1 is:

  • A circle whose center is 1.
  •  A straight.
  • A circle has center 1 except for a point.
  • A Straight excluding point.
b) The set of points M with a complex number Z where Z' is a real number is:
  • A circle whose center is 1.
  •  A straight.
  • A circle has center 1 except for a point.
  • A Straight except for a point.

Solution of the example :


1) Points  A, B , C are given their complex numbers, respectively  a-2+3i , b-3-i  , c2.08+1.98i .
AB = |b-a| 
AB = |-3-i+2-3i
AB = |-1-4i|
AB = (-1)²+(-4)² 

AB = 17 

AC =  |c-a|
AC =  |2.08+1.98i+2-3i|
AC =  |4.08-1.11i|
AC =  (4.08)²+(-1.11 

AC =  17 .8785

CB   |b-c|
CB =  |-3-i-2.08-1.98i|
CB =  |-5.08-2.98i|
CB =  (-5.08)²+(-2.98

CB =  34 .6868

Then from the measurements AB , AC , CB we conclude that:
  • The triangle is not right and not isosceles. ✔️

2)
a) For each complex number Z ≠ -2 we attach the complex number Z' where:
Z′=(Z−4i)/(Z+2)


The group of points M of complex numbers Z where |Z'|=1 is:

For every complex number   Z ≠ -2 : 

|Z'|=1  
 then: 

Z4iZ+2i=1     

 |Z-4i| = |Z+2|

Then:  AM = BM where : A(4i) and B(-2)

The set of points M with a complex number Z where: |Z'|=1 is:
  • A Straight except for the point B(-2) . ✔️
b) Then The set of points M with a complex number Z where Z' is a real number is

We write in the algebraic form by put: 

Z=x+iy

For every complex number   Z ≠ -2 : 

Z=Z4iZ+2(x+iy4i)x+iy+2


Z=(x+iy4i(x+2iy)(x+2+iy(x+2iy)


Z=x2+2xixy+ixy+i2y+y24ix8i4y(x+2)2+y2

Z=x2+y2+2x4y(x+2)2+y2+i4x+2y8(x+2)+y2

Z' is a real number this means :

4x+2y8(x+2)2+y2=0

This means :  -4x + 2y - 8=0 where ≠ -2 and  ≠ 0

Then The set of points M with a complex number Z where Z' is a real number is:
  • A Straight except for the point B(-2) . ✔️








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