Complex Numbers Exercise with Answer | Example 1

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 :

Δ=  (-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:

Δ= 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-i 

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= | Z_B -  Z_A| = |1+√3+i - 1-2i| = |√3-i| √ (√ 3)² +(-1)²  = 2

AC= | Z_c -  Z_A| = |1-2i - 1-2i| = |-4i| = √ (-4)²  = 4

CB= | Z_B -  Z_C| = |1+√3+i - 1-√3+i | = |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.

MA−→−⋅MC−→−=0

MA−→−=(1−x−2−y),MC−→−=(1−x2−y)

MA−→−⋅MC−→−=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: 

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

Read also:

complex numbers exercise with answer  | example 2

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