If `|z_1|` and `|z_2|` are the distance of points on the curve `5zbarz-2i(z^2-barz^2)-9=0` which are at maximum and minimum distance from the origin, then the value of `|z_1|+|z_2|` is equal to :

Answer

To solve the problem, we need to analyze the curve given by the equation:

\[ 5z \bar{z} - 2i(z^2 - \bar{z}^2) - 9 = 0 \]

### Step 1: Rewrite the equation using polar coordinates
Let \( z = re^{i\theta} \), where \( r = |z| \) (the modulus of \( z \)) and \( \theta \) is the argument of \( z \). Then, we have:

- \( \bar{z} = re^{-i\theta} \)
- \( z \bar{z} = r^2 \)
- \( z^2 = r^2 e^{2i\theta} \)
- \( \bar{z}^2 = r^2 e^{-2i\theta} \)

Substituting these into the equation gives:

\[ 5r^2 - 2i(r^2 e^{2i\theta} - r^2 e^{-2i\theta}) - 9 = 0 \]

### Step 2: Simplify the equation
The term \( e^{2i\theta} - e^{-2i\theta} \) can be simplified using the sine function:

\[ e^{2i\theta} - e^{-2i\theta} = 2i \sin(2\theta) \]

Thus, the equation becomes:

\[ 5r^2 - 2i(-2ir^2 \sin(2\theta)) - 9 = 0 \]

This simplifies to:

\[ 5r^2 + 4r^2 \sin(2\theta) - 9 = 0 \]

### Step 3: Rearranging the equation
Rearranging gives:

\[ 5r^2 + 4r^2 \sin(2\theta) = 9 \]

Factoring out \( r^2 \):

\[ r^2(5 + 4 \sin(2\theta)) = 9 \]

### Step 4: Solve for \( r^2 \)
Now, we can express \( r^2 \):

\[ r^2 = \frac{9}{5 + 4 \sin(2\theta)} \]

### Step 5: Determine the maximum and minimum values of \( r \)
To find the maximum and minimum values of \( r \), we need to analyze the expression \( 5 + 4 \sin(2\theta) \).

The sine function \( \sin(2\theta) \) varies between -1 and 1. Therefore:

- Maximum value of \( 5 + 4 \sin(2\theta) \) occurs when \( \sin(2\theta) = 1 \):
\[ 5 + 4(1) = 9 \]

- Minimum value of \( 5 + 4 \sin(2\theta) \) occurs when \( \sin(2\theta) = -1 \):
\[ 5 + 4(-1) = 1 \]

### Step 6: Calculate \( r^2 \) for maximum and minimum
1. For maximum \( 5 + 4 \sin(2\theta) = 9 \):
\[ r^2 = \frac{9}{9} = 1 \]
\[ r = 1 \]

2. For minimum \( 5 + 4 \sin(2\theta) = 1 \):
\[ r^2 = \frac{9}{1} = 9 \]
\[ r = 3 \]

### Step 7: Find \( |z_1| + |z_2| \)
Now, we have:

- \( |z_1| = 1 \) (minimum distance)
- \( |z_2| = 3 \) (maximum distance)

Thus, the value of \( |z_1| + |z_2| \) is:

\[ |z_1| + |z_2| = 1 + 3 = 4 \]

### Final Answer
The value of \( |z_1| + |z_2| \) is \( \boxed{4} \).

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