# 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, 190°.

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.

## 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, 190°.

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

z = (190°)6 = 1540° = 1180°

## 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) · 190° = (2 + i) · i = −1 + 2i = (−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.

(2270°)4 = 161080º = 163 · 360° = 16

## 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 + bi = rα

z = a − bi = r−α

r + r = 10 r = 5

a + a = 6 a = 3

52 = 32 + b2        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 + 4i = 553° 7' 48''

3 − 4i = 5306° 52' 11''