If the area of the triangle on the complex plane formed by the points z, iz and z+iz is 50 square units, then `|z|` is

Options

A. 5
B. 10
C. 15
D. none of these

Explanation

To solve the problem, we need to find the modulus of the complex number \( z \) given that the area of the triangle formed by the points \( z \), \( iz \), and \( z + iz \) is 50 square units.

### Step-by-Step Solution:

1. **Express \( z \) in terms of its components:**
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

2. **Identify the points on the complex plane:**
- The point \( z \) is \( (x, y) \).
- The point \( iz \) is \( (0, x) \) because \( iz = i(x + iy) = -y + ix \).
- The point \( z + iz \) is \( (x - y, x + y) \).

3. **Set up the vertices of the triangle:**
The vertices of the triangle in the coordinate system are:
- \( A(x, y) \)
- \( B(0, x) \)
- \( C(x - y, x + y) \)

4. **Use the formula for the area of a triangle given by three points:**
The area \( A \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
\[ A = \frac{1}{2} \left| x(x - (x + y)) + 0((x + y) - y) + (x - y)(y - x) \right| \]
Simplifying this:
\[ A = \frac{1}{2} \left| x(-y) + (x - y)(y - x) \right| \]
\[ = \frac{1}{2} \left| -xy + (x - y)(y - x) \right| \]
\[ = \frac{1}{2} \left| -xy + (x^2 - xy - yx + y^2) \right| \]
\[ = \frac{1}{2} \left| x^2 - 2xy + y^2 \right| \]
\[ = \frac{1}{2} \left| (x - y)^2 \right| \]

5. **Set the area equal to 50:**
Given that the area is 50 square units:
\[ \frac{1}{2} |(x - y)^2| = 50 \]
\[ |(x - y)^2| = 100 \]
\[ (x - y)^2 = 100 \]
\[ |x - y| = 10 \]

6. **Relate \( |z| \) to \( x \) and \( y \):**
The modulus of \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
We can express \( x \) in terms of \( y \) using \( |x - y| = 10 \):
- Case 1: \( x - y = 10 \) → \( x = y + 10 \)
- Case 2: \( y - x = 10 \) → \( x = y - 10 \)

7. **Calculate \( |z| \) for both cases:**
- **Case 1:** \( x = y + 10 \)
\[ |z| = \sqrt{(y + 10)^2 + y^2} = \sqrt{y^2 + 20y + 100 + y^2} = \sqrt{2y^2 + 20y + 100} \]
- **Case 2:** \( x = y - 10 \)
\[ |z| = \sqrt{(y - 10)^2 + y^2} = \sqrt{y^2 - 20y + 100 + y^2} = \sqrt{2y^2 - 20y + 100} \]

8. **Find the value of \( |z| \):**
Since we know \( |x - y| = 10 \), we can conclude that:
\[ |z| = 10 \]

### Final Answer:
Thus, the modulus of \( z \) is \( |z| = 10 \).