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 = √17
AC = |c-a|
AC = |2.08+1.98i+2-3i|
AC = |4.08-1.11i|
AC = √(4.08)²+(-1.11)²
AC = √17 .8785
CB = |b-c|
CB = |-3-i-2.08-1.98i|
CB = |-5.08-2.98i|
CB = √(-5.08)²+(-2.98)²
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:
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) . ✔️