**Complex numbers problems with solutions**

**Complex numbers exercise : **

Solve the following two equations in the set of complex numbers:

** Z²- 2Z + 5 =0 **

** Z² - 2(1+√3)Z + 5 + 2√3 = 0**

in Orthogonal coordinates * (o;i⇀,j⇀)* We consider the points A , B , C and D are the images of complex numbers

*1+2i , 1+*

*√3*

*+i ,*

*1-2i , 1+*

*√3*

*-i*respectively.

a) What is the nature of the triangle ABC ?

b) Write the equation of the circle *C* around the triangle ABC.

c) Prove that the point D belongs to the circle *C.*

d) Create C And the points A, B, C and D in the Orthogonal coordinates.

**solution of the example :**

Solve the following two equations in the set of complex numbers

** Z ²- 2Z + 5 =0 **

* Z² - 2(1+√3)Z + 5 + 2√3 = 0*

**A) ***Z ²- 2Z + 5 =0 *

Calculate the discriminant :

*Î”= (-2)²-4(5) = - 16 = (4i)² *

*Z _{1} = ^{(2-4i)}/_{2} = 1+2i , Z_{2} = ^{(2+4i)}/_{2} = 1+2i*

*B) Z² - 2(1+√3)Z + 5 + 2√3 = 0*

*Î”= 4(*

*1+√3)²-4(5+2*

*√3) = 4(1+2*

*√3)-20-8*

*√3 = -4 =*

*(2i)²*

*Z*

_{1}=^{[}

^{2}

^{(1+✓3)-2i]}/_{2}= 1+*√3-*

*i*

*Z _{2} = ^{[}*

^{2}

^{(1+✓3)+2i]}/_{2}= 1+*√3+*

*i*

#### a) The nature of the triangle ABC:

*1+2i ,*

**1+****√3+****i ,**1-2i ,

**1+****√3+****i***respectively*

*|*

*Z*

_{B}

*-*

*Z*

_{A}

*|*=

*|*

*1+√3+i*

*-*

*1-2i*

*| =*

*|*

*√3*

*-i*

*|*√ (

*√*3)² +(-1)² = 2

*|*

*Z*

_{c}

*-*

*Z*

_{A}

*|*=

*|*

*1-2i*

*-*

*1-2i*

*| =*

*|*

*-4i*

*| =*√ (-4)²

*= 4*

*|*

*Z*

_{B}

*-*

*Z*

_{C}

*|*=

*|*

*1+√3+i*

*-*

*1-*

*√3+i*

*| =*

*|*

*2i*

*|*√ 2² = 2

#### b) The equation of the circle *C* around the triangle ABC.

M (x;y) ∈ C this means.

**MA−→−⋅MC−→−=0**

**MA−→−=(1−x−2−y),MC−→−=(1−x2−y)**

**MA−→−⋅MC−→−=0 **

this means: *(1-x) (1-x) + (2-y) (-**2-y)=0*

*1-x-x+x²-4-2y+2y+y²=0*

**x²+y²-2x-3=0**

#### c) Prove that the point D belongs to the circle *C.*

*(1+√3)²+(-1)² - 2(1+√3) -3 = 1+3+2√3 +1-2-2√3-3**=0*

then the point D belongs to the circle *C .*

d) Create C And the points A, B, C and D in the Orthogonal coordinates.

**objectives of the exercises:**### solved problems on complex numbers

- Solve quadratic equations in complex numbers using the discriminant

- Determine the type of triangle using complex numbers

- Determine the equation of a circle that includes three points that belong to a triangle

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