The area of the triangle formed by the points representing `-z,iz` and `z-iz` in the Argand plane, is

Options

A. `1/2|z|^(2)`
B. `|z|^(2)`
C. `3/2|z|^(2)`
D. `1/4|z|^(2)`

Answer

To find the area of the triangle formed by the points representing \(-z\), \(iz\), and \(z - iz\) in the Argand plane, we can follow these steps:

### Step 1: Represent the complex number \(z\)
Let \(z = a + ib\), where \(a\) and \(b\) are real numbers.

### Step 2: Identify the points in the Argand plane
- The point representing \(-z\) is \(-a - ib\).
- The point representing \(iz\) is \(i(a + ib) = -b + ia\) (since \(i^2 = -1\)).
- The point representing \(z - iz\) is:
\[ z - iz = (a + ib) - (-b + ia) = (a + b) + i(b - a) \]
Thus, the coordinates of the points are:
- Point 1: \((-a, -b)\)
- Point 2: \((-b, a)\)
- Point 3: \((a + b, b - a)\)

### Step 3: Set up the area formula for the triangle
The area \(A\) of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 4: Substitute the points into the area formula
Substituting the coordinates of our points:
\[ A = \frac{1}{2} \left| (-a)(a - (b - a)) + (-b)((b - a) - (-b)) + (a + b)(-b - (-a)) \right| \]

### Step 5: Simplify the expression
Calculating each term:
1. First term:
\[ -a(a - (b - a)) = -a(2a - b) = -2a^2 + ab \]
2. Second term:
\[ -b((b - a) + b) = -b(2b - a) = -2b^2 + ab \]
3. Third term:
\[ (a + b)(-b + a) = (a + b)(a - b) = a^2 - b^2 + ab - ab = a^2 - b^2 \]

Combining these:
\[ A = \frac{1}{2} \left| (-2a^2 + ab) + (-2b^2 + ab) + (a^2 - b^2) \right| \]
\[ = \frac{1}{2} \left| -2a^2 - 2b^2 + 2ab + a^2 - b^2 \right| \]
\[ = \frac{1}{2} \left| -a^2 - 3b^2 + 2ab \right| \]

### Step 6: Express in terms of \(|z|^2\)
Recall that \(|z|^2 = a^2 + b^2\). Thus, we can rewrite:
\[ A = \frac{1}{2} \left| 3(a^2 + b^2) \right| = \frac{3}{2} |z|^2 \]

### Final Result
Thus, the area of the triangle formed by the points \(-z\), \(iz\), and \(z - iz\) is:
\[ \frac{3}{2} |z|^2 \]