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 will analyze the given quadratic equation and the conditions for the vertices of an equilateral triangle in the Argand plane.

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

To find the roots, we can use the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3 \), \( b = 3 \), and \( c = \lambda \).

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

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 roots
Using the quadratic formula:
\[ z = \frac{-3 \pm \sqrt{9 - 12\lambda}}{6} \]
Since the roots are non-real, we express them as:
\[ z_1 = \frac{-3 + i\sqrt{12\lambda - 9}}{6}, \quad z_2 = \frac{-3 - i\sqrt{12\lambda - 9}}{6} \]

### Step 4: Calculate the length of the triangle
The length of the triangle can be calculated as the distance between the two non-real roots \( z_1 \) and \( z_2 \):
\[ |z_1 - z_2| = | \left( \frac{-3 + i\sqrt{12\lambda - 9}}{6} \right) - \left( \frac{-3 - i\sqrt{12\lambda - 9}}{6} \right) | \]
This simplifies to:
\[ |z_1 - z_2| = | \frac{2i\sqrt{12\lambda - 9}}{6} | = \frac{|\sqrt{12\lambda - 9}|}{3} \]

### Step 5: Find the length of the equilateral triangle
In an equilateral triangle, the length of each side is the same. The distance from the origin to either of the roots is:
\[ |z_1| = |z_2| = \sqrt{\left(\frac{-3}{6}\right)^2 + \left(\frac{\sqrt{12\lambda - 9}}{6}\right)^2} = \sqrt{\frac{9}{36} + \frac{12\lambda - 9}{36}} = \sqrt{\frac{12\lambda}{36}} = \frac{\sqrt{12\lambda}}{6} \]

### Step 6: Calculate \( \sqrt{3} \) times the length of the triangle
The length of the triangle is \( |z_1 - z_2| \) which we calculated as \( \frac{|\sqrt{12\lambda - 9}|}{3} \). Therefore, \( \sqrt{3} \) times the length is:
\[ \sqrt{3} \cdot \frac{|\sqrt{12\lambda - 9}|}{3} = \frac{\sqrt{3}|\sqrt{12\lambda - 9}|}{3} \]

### Final Expression
Thus, the final expression for \( \sqrt{3} \) times the length of the triangle is:
\[ \frac{\sqrt{3}|\sqrt{12\lambda - 9}|}{3} \]