The roots of the cubic equation `(z + ab)^(3) = a ^(3) , a ne 0` represent the vertices of a triangle of sides of length

Options

A. `(1)/(sqrt(3)) |ab|`
B. `sqrt(3)|a|`
C. `sqrt(3)|b|`
D. `(1)/( sqrt(3))|a|`

Answer

To solve the given problem, we start with the cubic equation:

\[ (z + ab)^3 = a^3 \]

### Step 1: Simplify the Equation
We can rewrite the equation as:

\[ z + ab = a \cdot \omega^k \]

where \( \omega = e^{2\pi i / 3} \) is a primitive cube root of unity, and \( k = 0, 1, 2 \). This gives us three roots:

\[ z_1 = a - ab \]
\[ z_2 = a \cdot \omega - ab \]
\[ z_3 = a \cdot \omega^2 - ab \]

### Step 2: Calculate the Length of the Sides of the Triangle
We need to find the lengths of the sides of the triangle formed by these roots. The lengths of the sides can be calculated using the distance formula for complex numbers:

\[ |z_i - z_j| = |(x_1 + iy_1) - (x_2 + iy_2)| = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]

#### Length of Side \( z_1 - z_2 \)
Calculating \( |z_1 - z_2| \):

\[ z_1 - z_2 = (a - ab) - (a \cdot \omega - ab) = a(1 - \omega) \]

Now, we find the magnitude:

\[ |z_1 - z_2| = |a| \cdot |1 - \omega| \]

Using \( \omega = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \):

\[ 1 - \omega = 1 + \frac{1}{2} - i \frac{\sqrt{3}}{2} = \frac{3}{2} - i \frac{\sqrt{3}}{2} \]

Calculating the magnitude:

\[ |1 - \omega| = \sqrt{\left(\frac{3}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} \]

Thus,

\[ |z_1 - z_2| = |a| \cdot \sqrt{3} \]

#### Length of Side \( z_2 - z_3 \)
Calculating \( |z_2 - z_3| \):

\[ z_2 - z_3 = (a \cdot \omega - ab) - (a \cdot \omega^2 - ab) = a(\omega - \omega^2) \]

Calculating the magnitude:

\[ |z_2 - z_3| = |a| \cdot |\omega - \omega^2| \]

Using \( \omega - \omega^2 = \omega + \frac{1}{2} - i \frac{\sqrt{3}}{2} \):

\[ |\omega - \omega^2| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

Thus,

\[ |z_2 - z_3| = |a| \cdot 1 = |a| \]

#### Length of Side \( z_1 - z_3 \)
Calculating \( |z_1 - z_3| \):

\[ z_1 - z_3 = (a - ab) - (a \cdot \omega^2 - ab) = a(1 - \omega^2) \]

Calculating the magnitude:

\[ |z_1 - z_3| = |a| \cdot |1 - \omega^2| \]

Using \( 1 - \omega^2 = 1 + \frac{1}{2} + i \frac{\sqrt{3}}{2} = \frac{3}{2} + i \frac{\sqrt{3}}{2} \):

\[ |1 - \omega^2| = \sqrt{\left(\frac{3}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3} \]

Thus,

\[ |z_1 - z_3| = |a| \cdot \sqrt{3} \]

### Conclusion
The lengths of the sides of the triangle are:

1. \( |z_1 - z_2| = |a| \cdot \sqrt{3} \)
2. \( |z_2 - z_3| = |a| \cdot 1 \)
3. \( |z_1 - z_3| = |a| \cdot \sqrt{3} \)

Since all sides are equal, the triangle is equilateral.