If `z=(3+7i)(a+ib)`, where `a`, `b in Z-{0}`, is purely imaginery, then minimum value of `|z|^(2)` is

Options

A. `74`
B. `45`
C. `65`
D. `58`

Explanation

To solve the problem, we need to find the minimum value of \( |z|^2 \) given that \( z = (3 + 7i)(a + ib) \) and that \( z \) is purely imaginary. Let's break down the solution step by step.

### Step 1: Expand the expression for \( z \)
We start with the expression:
\[ z = (3 + 7i)(a + ib) \]
Expanding this using the distributive property:
\[ z = 3a + 3ib + 7ai + 7i^2b \]
Since \( i^2 = -1 \), we can substitute:
\[ z = 3a + 3ib + 7ai - 7b \]
Combining the real and imaginary parts:
\[ z = (3a - 7b) + i(3b + 7a) \]

### Step 2: Set the real part to zero
Since \( z \) is purely imaginary, the real part must equal zero:
\[ 3a - 7b = 0 \]
From this equation, we can express \( a \) in terms of \( b \):
\[ 3a = 7b \quad \Rightarrow \quad a = \frac{7b}{3} \]

### Step 3: Substitute \( a \) into the imaginary part
Now, we substitute \( a \) back into the imaginary part:
\[ \text{Imaginary part} = 3b + 7a = 3b + 7\left(\frac{7b}{3}\right) \]
Simplifying this:
\[ = 3b + \frac{49b}{3} = \frac{9b}{3} + \frac{49b}{3} = \frac{58b}{3} \]
Thus, we have:
\[ z = 0 + i\left(\frac{58b}{3}\right) = i\frac{58b}{3} \]

### Step 4: Calculate \( |z|^2 \)
The modulus squared of \( z \) is given by:
\[ |z|^2 = \left|\frac{58b}{3}\right|^2 = \left(\frac{58b}{3}\right)^2 = \frac{3364b^2}{9} \]

### Step 5: Minimize \( |z|^2 \)
To find the minimum value of \( |z|^2 \), we note that \( b \) must be a non-zero integer (as \( b \in \mathbb{Z} - \{0\} \)). The smallest absolute value for \( b \) is 1. Thus, substituting \( b = 1 \):
\[ |z|^2 = \frac{3364 \cdot 1^2}{9} = \frac{3364}{9} \]

### Conclusion
The minimum value of \( |z|^2 \) is:
\[ \frac{3364}{9} \]