`z=-1+sqrt3i , omega ` is complex number such that `abs(zomega)=1 , arg(z)-arg(w)=pi/2` .Find area of triangle made

Answer

To find the area of the triangle formed by the points \( z \), \( \omega \), and the origin, we will follow these steps:

### Step 1: Identify the given complex number \( z \)
Given:
\[ z = -1 + \sqrt{3}i \]

### Step 2: Calculate the modulus of \( z \)
The modulus of a complex number \( z = x + yi \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
For \( z = -1 + \sqrt{3}i \):
\[ |z| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 3: Find the complex number \( \omega \)
We know that \( |z \cdot \omega| = 1 \). Therefore:
\[ |z| \cdot |\omega| = 1 \implies |\omega| = \frac{1}{|z|} = \frac{1}{2} \]

### Step 4: Analyze the argument condition
We are given that:
\[ \arg(z) - \arg(\omega) = \frac{\pi}{2} \]
This implies that the angle between \( z \) and \( \omega \) is \( 90^\circ \). Therefore, \( \omega \) is perpendicular to \( z \).

### Step 5: Determine the position of \( \omega \)
Since \( z \) is in the second quadrant (as the real part is negative and the imaginary part is positive), \( \omega \) will be in the fourth quadrant due to the angle condition.

### Step 6: Calculate the coordinates of \( \omega \)
Assuming \( \omega = x + yi \) and knowing \( |\omega| = \frac{1}{2} \):
\[ \sqrt{x^2 + y^2} = \frac{1}{2} \implies x^2 + y^2 = \frac{1}{4} \]
Given that \( \arg(z) - \arg(\omega) = \frac{\pi}{2} \), we can express \( \omega \) in terms of \( z \):
\[ \omega = k(-\sqrt{3} - i) \quad \text{for some scalar } k \]
To satisfy \( |\omega| = \frac{1}{2} \), we can find \( k \) such that:
\[ |k| \cdot \sqrt{(-\sqrt{3})^2 + (-1)^2} = \frac{1}{2} \]
Calculating the modulus:
\[ \sqrt{3 + 1} = 2 \implies |k| \cdot 2 = \frac{1}{2} \implies |k| = \frac{1}{4} \]
Thus, we can take \( k = \frac{1}{4} \):
\[ \omega = \frac{1}{4}(-\sqrt{3} - i) = -\frac{\sqrt{3}}{4} - \frac{1}{4}i \]

### Step 7: Calculate the area of the triangle
The area \( A \) of a triangle formed by points \( (0, 0) \), \( (x_1, y_1) \), and \( (x_2, y_2) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1y_2 - x_2y_1 \right| \]
For our points:
- \( z = (-1, \sqrt{3}) \)
- \( \omega = \left(-\frac{\sqrt{3}}{4}, -\frac{1}{4}\right) \)

Calculating the area:
\[ A = \frac{1}{2} \left| (-1)(-\frac{1}{4}) - \left(-\frac{\sqrt{3}}{4}\right)(\sqrt{3}) \right| \]
\[ = \frac{1}{2} \left| \frac{1}{4} + \frac{3}{4} \right| = \frac{1}{2} \left| 1 \right| = \frac{1}{2} \]

### Final Answer
The area of the triangle formed by the points \( z \), \( \omega \), and the origin is:
\[ \boxed{\frac{1}{2}} \]