Let `z=x+i y` be a complex number where `xa n dy` are integers. Then, the area of the rectangle whose vertices are the roots of the equation `z z ^3+ z z^3=350` is 48 (b) 32 (c) 40 (d) 80

Options

A. 48
B. 32
C. 40
D. 80

Explanation

To solve the problem step by step, we will analyze the equation given and find the area of the rectangle formed by the roots.

### Step 1: Understand the given equation
The equation we have is:
\[ z + \bar{z}^3 + z \bar{z}^3 = 350 \]
where \( z = x + iy \) and \( \bar{z} = x - iy \).

### Step 2: Rewrite the equation
We can rewrite the equation using the properties of complex numbers:
\[ z + \bar{z}^3 + z \bar{z}^3 = 350 \]

### Step 3: Substitute \( z \) and \( \bar{z} \)
Substituting \( z = x + iy \) and \( \bar{z} = x - iy \):
1. Calculate \( \bar{z}^3 \):
\[ \bar{z}^3 = (x - iy)^3 = x^3 - 3x^2(iy) + 3x(iy)^2 - (iy)^3 = x^3 - 3x^2(iy) - 3xy^2 - iy^3 \]
Simplifying gives:
\[ \bar{z}^3 = (x^3 - 3xy^2) - i(3x^2y - y^3) \]

2. Calculate \( z \bar{z}^3 \):
\[ z \bar{z}^3 = (x + iy)((x^3 - 3xy^2) - i(3x^2y - y^3)) \]

### Step 4: Combine terms
Now, we will combine the terms and set the real part equal to 350. However, for simplicity, we can analyze the equation in terms of magnitudes:
\[ z + \bar{z}^3 + z \bar{z}^3 = 350 \]

### Step 5: Factor the equation
We can factor the equation as follows:
\[ z^2(z + \bar{z}) = 350 \]
This implies:
\[ z^2 \cdot (x + iy + x - iy) = 350 \]
Thus:
\[ z^2 \cdot 2x = 350 \]

### Step 6: Set up equations
Let:
\[ x^2 + y^2 = A \quad \text{and} \quad x^2 - y^2 = B \]
From the equation \( A \cdot B = 175 \).

### Step 7: Solve for integer values
We can find pairs of integers \( (A, B) \) such that:
\[ A \cdot B = 175 \]
The factor pairs of 175 are:
- \( (1, 175) \)
- \( (5, 35) \)
- \( (7, 25) \)

### Step 8: Check pairs for integer solutions
1. For \( A = 25 \) and \( B = 7 \):
\[ x^2 + y^2 = 25 \quad \text{and} \quad x^2 - y^2 = 7 \]
Adding these gives:
\[ 2x^2 = 32 \implies x^2 = 16 \implies x = \pm 4 \]
Substituting back gives:
\[ y^2 = 9 \implies y = \pm 3 \]

### Step 9: Determine the vertices
The vertices of the rectangle formed by the roots are:
- \( (4, 3) \)
- \( (4, -3) \)
- \( (-4, 3) \)
- \( (-4, -3) \)

### Step 10: Calculate the area of the rectangle
The lengths of the sides of the rectangle are:
- Length = \( 8 \) (from \( -4 \) to \( 4 \))
- Width = \( 6 \) (from \( -3 \) to \( 3 \))

Thus, the area \( A \) is:
\[ A = \text{Length} \times \text{Width} = 8 \times 6 = 48 \]

### Final Answer
The area of the rectangle is \( 48 \).

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