# Complex Numbers Exercise with Answer | Example 1

## 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

Z1 = (2-4i)/2    = 1+2i    ,    Z2 = (2+4i)/2    = 1+2i

B) Z² - 2(1+√3)Z + 5 + 2√3 = 0

Calculate the  discriminant:

Î”= 4(1+√3)²-4(5+2√3) = 4(1+2√3)-20-8√3 = -4 =(2i)²

Z1 = [2(1+✓3)-2i]/2  = 1+√3-

Z2 = [2(1+✓3)+2i]/2  = 1+√3+i

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

We consider points A ,B , C , D images of complex numbers 1+2i , 1+√3+i  , 1-2i , 1+√3+i  respectively

AB= | ZB -  ZA||1+√3+1-2i| = |√3-i ( 3)² +(-1)²  = 2

AC= | Zc -  ZA| = |1-2i 1-2i| = |-4i| =  (-4)²  = 4

CB= | ZB -  ZC| = |1+√3+1-√3+| = |2i 2² = 2

since: AB²+CB²=AC² then the triangle ABC is a right triangle according to the Pythagoras Theorem.

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

M (x;y) ∈ C this means.

MAMC=0

MA=(1x2y),MC=(1x2y)

MAMC=0    i

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: