Let `|z_1|=i,i=1,2,3,4and |16 z_1z_2z_3+9z_1z_2z_4+4z_1z_3z_4+z_2z_3z_4|=48` ,then the value of `|1/barz_1+4/bar z_2+9/vec z_3+16/bar z_4|`

Options

A. 1
B. 2
C. 4
D. 8

Explanation

To solve the problem step by step, we will analyze the given conditions and derive the required expression.

### Step 1: Understand the given information
We know that:
- \(|z_1| = 1\)
- \(|z_2| = 2\)
- \(|z_3| = 3\)
- \(|z_4| = 4\)

We also have the equation:
\[ |16 z_1 z_2 z_3 + 9 z_1 z_2 z_4 + 4 z_1 z_3 z_4 + z_2 z_3 z_4| = 48 \]

### Step 2: Factor out the modulus
Using the property of moduli, we can factor out \(|z_1 z_2 z_3 z_4|\) from the expression:
\[ |z_1 z_2 z_3 z_4| \cdot |16 \frac{1}{z_4} + 9 \frac{1}{z_3} + 4 \frac{1}{z_2} + \frac{1}{z_1}| = 48 \]

### Step 3: Calculate \(|z_1 z_2 z_3 z_4|\)
We calculate:
\[ |z_1 z_2 z_3 z_4| = |z_1| \cdot |z_2| \cdot |z_3| \cdot |z_4| = 1 \cdot 2 \cdot 3 \cdot 4 = 24 \]

### Step 4: Substitute back into the equation
Now substituting back into the equation:
\[ 24 \cdot |16 \frac{1}{z_4} + 9 \frac{1}{z_3} + 4 \frac{1}{z_2} + \frac{1}{z_1}| = 48 \]

### Step 5: Solve for the modulus expression
Dividing both sides by 24 gives:
\[ |16 \frac{1}{z_4} + 9 \frac{1}{z_3} + 4 \frac{1}{z_2} + \frac{1}{z_1}| = \frac{48}{24} = 2 \]

### Final Result
Thus, we find that:
\[ |1/z_1 + 4/z_2 + 9/z_3 + 16/z_4| = 2 \]

### Conclusion
The value of \(|1/\bar{z_1} + 4/\bar{z_2} + 9/\bar{z_3} + 16/\bar{z_4}|\) is \(2\).

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