A complex number `z` satisfies the equation `|Z^(2)-9|+|Z^(2)|=41`, then the true statements among the following are

Options

A. `|Z+3|+|Z-3|=10`
B. `|Z+3|+|Z-3|=8`
C. Maximum value of `|Z|` is `5`
D. Maximum value of `|Z|` is `6`

Explanation

To solve the equation \( |z^2 - 9| + |z^2| = 41 \) for the complex number \( z \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ |z^2 - 9| + |z^2| = 41 \]

### Step 2: Use the triangle inequality
We can apply the triangle inequality:
\[ |z^2 - 9| + |z^2| \geq |(z^2 - 9) + z^2| = |2z^2 - 9| \]
This means:
\[ |2z^2 - 9| \leq 41 \]

### Step 3: Analyze the expression
From the triangle inequality, we can derive:
\[ -41 \leq 2z^2 - 9 \leq 41 \]
This implies:
\[ -41 + 9 \leq 2z^2 \leq 41 + 9 \]
\[ -32 \leq 2z^2 \leq 50 \]

### Step 4: Divide by 2
Dividing the entire inequality by 2 gives:
\[ -16 \leq z^2 \leq 25 \]

### Step 5: Consider the modulus
Since \( z^2 \) is a complex number, we can consider the modulus:
\[ |z^2| \leq 25 \]
This means:
\[ |z|^2 \leq 25 \implies |z| \leq 5 \]

### Step 6: Maximum value of \( |z| \)
The maximum value of \( |z| \) is therefore:
\[ |z| \leq 5 \]

### Step 7: Verify the conditions
We also need to check if the equality can hold. The equality \( |z^2 - 9| + |z^2| = 41 \) can be satisfied under certain conditions, but we have established the bounds for \( |z| \).

### Conclusion
The true statements among the options provided are:
1. \( |z| \leq 5 \)
2. The equation can hold under certain conditions.