# Complex Number Word Problems

### Solutions

1 Calculate the value of k for the complex number obtained by dividing . Note, it is represented in the bisector of the first quadrant.

2 Find the value of k for the quotient , if it is:

1 A pure imaginary number.

2 A real number.

3 The complex number, 2 + 2 * is rotated 45° about the origin of its coordinates in an anti-clockwise direction*. Find the complex number obtained after the turn.

4 Find the coordinates of the vertices of a regular hexagon of center origin, knowing that one of its vertices is the affix of complex number, 1_{90°}_{. }

5Determine the value of b for the quotient , if it equals:

6What are the coordinates of the point that is obtained on having turned the affix of the complex 2 + i, 90° in an counterclockwise direction about the origin.

7 Find the coordinates of the vertices of a square of center origin, knowing that one vertex is the point (0, −2).

8 The sum of the real components of two conjugate complex numbers is six, and the sum of its modulus is 10. Determine these complex numbers.

## Solved Complex Number Word Problems

## 1

Calculate the value of k for the complex number obtained by dividing . Note, it is represented in the bisector of the first quadrant.

For the affix, (a, b), the complex number is on the bisector of the first quadrant. The following must be met: a = b.

## 2

Find the value of k for the quotient , if it is:

1 A pure imaginary number.

2 A real number.

## 3

The complex number, 2 + 2 * is rotated 45° about the origin of its coordinates in an anti-clockwise direction*. Find the complex number obtained after the turn.

## 4

Find the coordinates of the vertices of a regular hexagon of center origin, knowing that one of its vertices is the affix of complex number, 1_{90°}_{. }

The vertices are the affixes of the sixth roots of another complex number, z.

**z = (1 _{90°})^{6 } = 1_{540° } = 1_{180° } **

## 5

Determine the value of b for the quotient

## 6

What are the coordinates of the point that is obtained on having turned the affix of the complex 2 + i, 90° in an counterclockwise direction about the origin.

(2 + *i*) · 1_{90°} = (2 + *i*) · *i* = −1 + 2*i* = (−1,2)

## 7

Find the coordinates of the vertices of a square of center origin, knowing that one vertex is the point (0, −2).

**(0, −2) = −2 i = 2_{ 270}**

The vertices are the affixes of the quarter roots of another complex number, z.

**(2 _{270°})^{4} = 16_{1080º} = 16_{3 · 360° } = 16_{0°}**

## 8

The sum of the real components of two conjugate complex numbers is six, and the sum of its modulus is 10. Determine these complex numbers.

**z = a + b i = r_{α}**

** z = a − b i = r_{−α}**

**r + r = 10 r = 5**

**a + a = 6 a = 3 **

**5 ^{2} = 3^{2} + b^{2}**

**b=4**

**r cos α + r cos (−α) = 6**

**5 cos α + 5 cos α = 6 **

**cos α = 3/5**

**α = 53° 7' 48'' α = 306° 52' 11''**

**3 + 4 i = 5_{53° 7' 48'' } **

**3 − 4 i = 5_{306° 52' 11''}**