if `A(z_1),B(z_2),C(z_3),D(z_4)` lies on |z|=4 (taken in order) , where `z_1+z_2+z_3+z_4=0` then :

Options

A. A) Max. area of quadrilateral ABCD=32
B. B) Max. area of quadrilateral ABCD=16
C. C) The triangle `DeltaABC` is right angled
D. D) The quadrilateral ABCD is rectangle

Answer

To solve the problem step by step, we need to analyze the given conditions and derive the maximum area of the quadrilateral formed by the points \( A(z_1), B(z_2), C(z_3), D(z_4) \) on the circle defined by \( |z| = 4 \).

### Step 1: Understand the given conditions
We know that the points \( A(z_1), B(z_2), C(z_3), D(z_4) \) lie on the circle defined by \( |z| = 4 \). This means that each point can be represented in the complex plane as:
\[ z_1 = 4e^{i\theta_1}, \quad z_2 = 4e^{i\theta_2}, \quad z_3 = 4e^{i\theta_3}, \quad z_4 = 4e^{i\theta_4} \]
for some angles \( \theta_1, \theta_2, \theta_3, \theta_4 \).

### Step 2: Use the condition \( z_1 + z_2 + z_3 + z_4 = 0 \)
The condition \( z_1 + z_2 + z_3 + z_4 = 0 \) implies that the center of mass of the points is at the origin. This suggests that the points are symmetrically distributed around the origin.

### Step 3: Determine the configuration for maximum area
The maximum area of a quadrilateral inscribed in a circle occurs when the quadrilateral is a square. Given that the radius of the circle is 4, we can place the vertices of the square at:
\[ z_1 = 4, \quad z_2 = 4i, \quad z_3 = -4, \quad z_4 = -4i \]

### Step 4: Calculate the area of the square
The side length of the square can be calculated as follows:
\[ \text{Side length} = \sqrt{(4 - (-4))^2 + (0 - 0)^2} = \sqrt{(8)^2} = 8 \]
The area \( A \) of the square is given by:
\[ A = \text{side}^2 = 8^2 = 64 \]

### Step 5: Conclusion
However, we need to ensure that the area calculated is indeed the maximum area of the quadrilateral. The maximum area of the quadrilateral formed by points on the circle can also be calculated using the formula for the area of a cyclic quadrilateral:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. For a square inscribed in a circle of radius \( r \):
\[ d_1 = d_2 = 8 \quad \text{(diagonal of the square)} \]
Thus, the area is:
\[ \text{Area} = \frac{1}{2} \times 8 \times 8 = 32 \]

### Final Answer
The maximum area of quadrilateral \( ABCD \) is \( 32 \).