**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 , *c*= **2.08+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.

**Z′=(Z−4i)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.

- 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 ,*c*=**2.08+1.98i .**AB = |b-a|*

*AB = |*

*-3-i+2-3i*|*AB = |*

*-1-4i*|*AB =*√(-1)²+(-4)²

*AB =*√17AC =

*|c-a|**AC =*

*|**2.08+1.98i**|**+2-3i*

*AC =**|4.08-1.11i|**√(*

*AC =**)²+(*

*4.08**)²*

*-1.11*

**AC =****√17 .8785**

CB =

*|b-c|*CB =

*|**-3-i*-*|**2.08-1.98i**CB =*

*|**-5.08*-*|**2.98i**)²*

*√(**CB =**)²+(**-5.08**-**2.98*

**CB =****√34 .6868**

Then from the measurements

**we conclude that:***AB , AC , CB*- The triangle is not right and not isosceles. ✔️

2)

a) For each complex number Z ≠ -2 we attach the complex number Z' where:

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

For every complex number Z ≠ -2 :

|Z'|=1

then:

**∣∣∣Z−4iZ+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′=Z−4iZ+2= (x+iy−4i)x+iy+2**

**Z′=(x+iy−4i) (x+2−iy)(x+2+iy) (x+2−iy)**

**Z′=x2+2x−ixy+ixy+i2y+y2−4ix−8i−4y(x+2)2+y2**

**Z′=x2+y2+2x−4y(x+2)2+y2+i−4x+2y−8(x+2)2 +y2**

*Z'*is a real number this means :

**−4x+2y−8(x+2)2+y2=0**

This means :

*-4x + 2y - 8=0*where*x**≠ -2*and*y*≠ 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*) . ✔️

## Comments

## Post a Comment