Area of the triangle in the argand diagram formed by the complex numbers z, iz, z +iz , where z = x+iy , is

Options

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

Answer

To find the area of the triangle formed by the complex numbers \( z \), \( iz \), and \( z + iz \) in the Argand diagram, where \( z = x + iy \), we can follow these steps:

### Step 1: Identify the Complex Numbers
Given:
- \( z = x + iy \)
- \( iz = i(x + iy) = ix - y = -y + ix \)
- \( z + iz = (x + iy) + (-y + ix) = (x - y) + (x + y)i \)

### Step 2: Represent the Complex Numbers in Cartesian Coordinates
The points in the Argand diagram corresponding to these complex numbers are:
- \( A(z) = (x, y) \)
- \( B(iz) = (-y, x) \)
- \( C(z + iz) = (x - y, x + y) \)

### Step 3: Use the Area Formula for a Triangle
The area \( A \) of a triangle formed by three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 4: Plug in the Coordinates
Substituting the coordinates of points \( A \), \( B \), and \( C \):
- \( (x_1, y_1) = (x, y) \)
- \( (x_2, y_2) = (-y, x) \)
- \( (x_3, y_3) = (x - y, x + y) \)

The area \( A \) becomes:
\[ A = \frac{1}{2} \left| x(x - (x + y)) + (-y)((x + y) - y) + (x - y)(y - x) \right| \]

### Step 5: Simplify the Expression
Calculating each term:
1. \( x(x - (x + y)) = x(x - x - y) = -xy \)
2. \( -y((x + y) - y) = -y(x) = -xy \)
3. \( (x - y)(y - x) = (x - y)(-x + y) = -(x - y)(x - y) = -(x - y)^2 \)

So, the area becomes:
\[ A = \frac{1}{2} \left| -xy - xy - (x - y)^2 \right| \]
\[ = \frac{1}{2} \left| -2xy - (x^2 - 2xy + y^2) \right| \]
\[ = \frac{1}{2} \left| -x^2 - y^2 \right| \]
\[ = \frac{1}{2} (x^2 + y^2) \]

### Step 6: Relate to the Modulus of \( z \)
Since \( |z|^2 = x^2 + y^2 \), we can write:
\[ A = \frac{1}{2} |z|^2 \]

### Final Answer
The area of the triangle formed by the complex numbers \( z \), \( iz \), and \( z + iz \) is:
\[ \text{Area} = \frac{1}{2} |z|^2 \]