If the origin and the non - real roots of the equation `3z^(2)+3z+lambda=0, AA lambda in R` are the vertices of an equilateral triangle in the argand plane, then `sqrt3` times the length of the triangle is

Options

A. 2 units
B. 1 units
C. 3 units
D. 4 units

Answer

To solve the problem step by step, we need to analyze the quadratic equation and the properties of the roots in relation to the Argand plane.

### Step 1: Identify the roots of the quadratic equation
The given quadratic equation is:
\[ 3z^2 + 3z + \lambda = 0 \]

The roots of this equation can be found using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = 3 \), and \( c = \lambda \).

### Step 2: Calculate the discriminant
The discriminant \( D \) of the quadratic equation is given by:
\[ D = b^2 - 4ac = 3^2 - 4 \cdot 3 \cdot \lambda = 9 - 12\lambda \]

For the roots to be non-real, the discriminant must be less than zero:
\[ 9 - 12\lambda < 0 \]
This simplifies to:
\[ \lambda > \frac{3}{4} \]

### Step 3: Find the sum and product of the roots
Using Vieta's formulas, we know:
- The sum of the roots \( z_1 + z_2 = -\frac{b}{a} = -1 \)
- The product of the roots \( z_1 z_2 = \frac{c}{a} = \frac{\lambda}{3} \)

### Step 4: Position of the roots in the Argand plane
Let the roots be \( z_1 \) and \( z_2 \), and the origin is at \( 0 \). The points \( z_1 \), \( z_2 \), and \( 0 \) form an equilateral triangle.

### Step 5: Use the properties of the equilateral triangle
In an equilateral triangle, the distance between any two vertices is the same. Let the length of each side of the triangle be \( L \). The distance from the origin to each root will also be \( L \).

### Step 6: Relationship of the roots
Since \( z_1 \) and \( z_2 \) are the vertices of the equilateral triangle with the origin, we can express the relationship between the roots using complex numbers:
\[ z_2 = z_1 e^{i\frac{\pi}{3}} \]
This means that the angle between the lines connecting the origin to the roots is \( 60^\circ \).

### Step 7: Calculate the length of the sides
Using the distance formula, we can express the length \( L \) in terms of the roots:
\[ |z_1 - z_2| = |z_1 - z_1 e^{i\frac{\pi}{3}}| = |z_1| |1 - e^{i\frac{\pi}{3}}| \]

The magnitude of \( 1 - e^{i\frac{\pi}{3}} \) can be calculated as:
\[ |1 - e^{i\frac{\pi}{3}}| = |1 - \left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right)| = |1 - \frac{1}{2} - i\frac{\sqrt{3}}{2}| = |\frac{1}{2} - i\frac{\sqrt{3}}{2}| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

### Step 8: Find the final length
Thus, the length \( L \) is:
\[ L = |z_1| \cdot 1 = |z_1| \]

### Step 9: Calculate \( \sqrt{3} \) times the length of the triangle
Finally, we need to find \( \sqrt{3} \) times the length of the triangle:
\[ \sqrt{3} \cdot L = \sqrt{3} \cdot 1 = \sqrt{3} \]

### Conclusion
The value of \( \sqrt{3} \) times the length of the triangle is:
\[ \boxed{1} \]