If z = x + iy and roots ` zbarz^(3) + barz z^(3) = 30` are the vertices of a rectangle and ` z_(0)` is centre of rectangle. Let d be distance of ` z_(0)` form the point on circle |z-3| `le ` 2 then maximum value of d is ________

Answer

To solve the problem step by step, we will analyze the given equation and the conditions provided.

### Step 1: Understand the given equation
We start with the equation:
\[ z \bar{z}^3 + \bar{z} z^3 = 30 \]
where \( z = x + iy \) and \( \bar{z} = x - iy \).

### Step 2: Rewrite the equation
We can factor out \( z \) and \( \bar{z} \):
\[ z \bar{z} (\bar{z}^2 + z^2) = 30 \]
Since \( z \bar{z} = |z|^2 = x^2 + y^2 \), we rewrite it as:
\[ |z|^2 (\bar{z}^2 + z^2) = 30 \]

### Step 3: Expand \( \bar{z}^2 + z^2 \)
Now, we can express \( \bar{z}^2 + z^2 \):
\[ \bar{z}^2 = (x - iy)^2 = x^2 - 2xyi - y^2 \]
\[ z^2 = (x + iy)^2 = x^2 + 2xyi - y^2 \]
Adding these gives:
\[ \bar{z}^2 + z^2 = 2(x^2 - y^2) \]
Thus, the equation becomes:
\[ |z|^2 \cdot 2(x^2 - y^2) = 30 \]

### Step 4: Substitute \( |z|^2 \)
Let \( r^2 = x^2 + y^2 \). Then we can rewrite the equation:
\[ r^2 \cdot 2(x^2 - y^2) = 30 \]
This implies:
\[ r^2 (x^2 - y^2) = 15 \]

### Step 5: Set up the equations
From the equation \( r^2 (x^2 - y^2) = 15 \), we can express:
1. \( x^2 + y^2 = r^2 \)
2. \( x^2 - y^2 = \frac{15}{r^2} \)

### Step 6: Solve for \( x^2 \) and \( y^2 \)
Adding and subtracting these two equations gives:
\[ 2x^2 = r^2 + \frac{15}{r^2} \implies x^2 = \frac{r^2 + \frac{15}{r^2}}{2} \]
\[ 2y^2 = r^2 - \frac{15}{r^2} \implies y^2 = \frac{r^2 - \frac{15}{r^2}}{2} \]

### Step 7: Find possible values
To find the possible values of \( x^2 \) and \( y^2 \), we need to ensure that both \( x^2 \) and \( y^2 \) are non-negative. This leads us to find suitable values of \( r^2 \).

### Step 8: Center of the rectangle
The center \( z_0 \) of the rectangle formed by the roots is at the origin \( (0, 0) \).

### Step 9: Distance from \( z_0 \) to the circle
The circle given is \( |z - 3| \leq 2 \), which has a center at \( (3, 0) \) and radius \( 2 \). The distance from the center of the rectangle \( z_0 \) to the center of the circle is:
\[ d = |3 - 0| = 3 \]

### Step 10: Maximum distance \( d \)
The maximum distance \( d \) from \( z_0 \) to any point on the circle is:
\[ d_{\text{max}} = 3 + 2 = 5 \]

### Final Answer
Thus, the maximum value of \( d \) is:
\[ \boxed{5} \]