Introduction:
We are given two complex numbers, z₁ and z₂, and we need to find the minimum value of |z₁ - z₂| based on the given conditions.

Given Information:
- |z₁| = 9
- |z₂ - 3 - 4i| = 4

Approach:
To find the minimum value of |z₁ - z₂|, we need to determine the possible values of z₁ and z₂ that satisfy the given conditions. By analyzing these values, we can find the minimum distance between z₁ and z₂.

Solution:

Step 1: Finding the Possible Values of z₁:
Since |z₁| = 9, we know that z₁ lies on a circle centered at the origin (0,0) with a radius of 9 units. Therefore, z₁ can be any complex number that lies on this circle.

Step 2: Finding the Possible Values of z₂:
We are given that |z₂ - 3 - 4i| = 4. This represents a circle centered at (3,4) with a radius of 4 units. Therefore, z₂ can be any complex number that lies on this circle.

Step 3: Determining the Minimum Distance:
To find the minimum distance between z₁ and z₂, we need to find the closest points on these circles. Since the circles are not concentric, the minimum distance will be the distance between the two closest points on the circles.

Step 4: Calculating the Distance:
Let's assume the closest point on the circle centered at the origin is A, and the closest point on the circle centered at (3,4) is B. The distance between A and B will be the minimum distance between z₁ and z₂.

Step 5: Finding the Coordinates of A and B:
To find the coordinates of A and B, we can use the concept of similar triangles. The triangle formed by the origin, A, and the center of the circle centered at (3,4) is similar to the triangle formed by A, B, and the center of the circle centered at (3,4). This similarity allows us to find the coordinates of A and B.

Step 6: Calculating the Minimum Distance:
Once we have the coordinates of A and B, we can calculate the distance between them using the distance formula. This distance will be the minimum value of |z₁ - z₂|.

Conclusion:
By following the above steps, we can find the minimum value of |z₁ - z₂| based on the given conditions.