If `lambda in R` such that the origin and the non-real roots of the equation `2z^(2)+2z+lambda=0` form the vertices of an equilateral triangle in the argand plane, then `(1)/(lambda)` is equal to

Answer

To solve the problem, we need to find the value of \( \frac{1}{\lambda} \) given that the origin and the non-real roots of the equation \( 2z^2 + 2z + \lambda = 0 \) form the vertices of an equilateral triangle in the Argand plane.

### Step 1: Identify the roots of the quadratic equation
The roots of the quadratic equation \( 2z^2 + 2z + \lambda = 0 \) can be found using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = 2 \), and \( c = \lambda \). Therefore, the roots are:
\[ z = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 2 \cdot \lambda}}{2 \cdot 2} = \frac{-2 \pm \sqrt{4 - 8\lambda}}{4} \]
For the roots to be non-real, the discriminant must be negative:
\[ 4 - 8\lambda < 0 \implies \lambda > \frac{1}{2} \]

### Step 2: Use the properties of the roots
Let the non-real roots be \( z_1 \) and \( z_2 \). By Vieta's formulas, we know:
\[ z_1 + z_2 = -\frac{b}{a} = -\frac{2}{2} = -1 \]
\[ z_1 z_2 = \frac{c}{a} = \frac{\lambda}{2} \]

### Step 3: Condition for equilateral triangle
For the points \( z_1, z_2, \) and the origin (0) to form an equilateral triangle, the following condition must hold:
\[ z_1^2 + z_2^2 = z_1 z_2 \]
We can express \( z_1^2 + z_2^2 \) in terms of \( z_1 + z_2 \) and \( z_1 z_2 \):
\[ z_1^2 + z_2^2 = (z_1 + z_2)^2 - 2z_1 z_2 \]
Substituting the values:
\[ z_1^2 + z_2^2 = (-1)^2 - 2\left(\frac{\lambda}{2}\right) = 1 - \lambda \]
Now, substituting this into the equilateral triangle condition:
\[ 1 - \lambda = \frac{\lambda}{2} \]

### Step 4: Solve for \( \lambda \)
Rearranging the equation:
\[ 1 - \lambda = \frac{\lambda}{2} \]
Multiplying through by 2 to eliminate the fraction:
\[ 2 - 2\lambda = \lambda \]
Bringing all terms involving \( \lambda \) to one side:
\[ 2 = 3\lambda \implies \lambda = \frac{2}{3} \]

### Step 5: Find \( \frac{1}{\lambda} \)
Now, we can find \( \frac{1}{\lambda} \):
\[ \frac{1}{\lambda} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

### Final Answer
Thus, the value of \( \frac{1}{\lambda} \) is:
\[ \frac{3}{2} \]