The locus of the centre of a circle which touches the given circles `|z -z_(1)| = |3 + 4i| and |z-z_(2)| =|1+isqrt3|` is a hyperbola, then the lenth of its transvers axis is ……

Answer

To solve the problem, we need to find the length of the transverse axis of the hyperbola formed by the locus of the center of a circle that touches two given circles. We will follow these steps:

### Step 1: Identify the centers and radii of the given circles

The first circle is given by the equation:
\[ |z - z_1| = |3 + 4i| \]
Here, the center \( z_1 \) is at the origin, and the radius \( r_1 \) is the magnitude of \( 3 + 4i \):
\[ r_1 = |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

The second circle is given by the equation:
\[ |z - z_2| = |1 + i\sqrt{3}| \]
Here, the center \( z_2 \) is at the point \( 1 + i\sqrt{3} \), and the radius \( r_2 \) is the magnitude of \( 1 + i\sqrt{3} \):
\[ r_2 = |1 + i\sqrt{3}| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 2: Set up the equations for the distances

Let \( z_0 \) be the center of the circle that touches both given circles. The conditions for tangency give us the following equations:

1. The distance from \( z_0 \) to \( z_1 \) is equal to the sum of the radius \( r_0 \) of the circle centered at \( z_0 \) and the radius \( r_1 \):
\[ |z_0 - z_1| = r_0 + r_1 = r_0 + 5 \]

2. The distance from \( z_0 \) to \( z_2 \) is equal to the sum of the radius \( r_0 \) and the radius \( r_2 \):
\[ |z_0 - z_2| = r_0 + r_2 = r_0 + 2 \]

### Step 3: Subtract the two equations

From the two equations above, we can express them as:
1. \( |z_0 - z_1| = r_0 + 5 \) (Equation 1)
2. \( |z_0 - z_2| = r_0 + 2 \) (Equation 2)

Now, subtract Equation 2 from Equation 1:
\[ |z_0 - z_1| - |z_0 - z_2| = (r_0 + 5) - (r_0 + 2) \]
This simplifies to:
\[ |z_0 - z_1| - |z_0 - z_2| = 3 \]

### Step 4: Identify the form of the hyperbola

The equation \( |z_0 - z_1| - |z_0 - z_2| = 3 \) represents a hyperbola, where the distance between the foci is \( 2c \) and the transverse axis is \( 2a \). The constant on the right side (3) represents \( 2a \).

### Step 5: Calculate the length of the transverse axis

From the equation \( |z_0 - z_1| - |z_0 - z_2| = 3 \), we can directly identify that:
\[ 2a = 3 \]
Thus, the length of the transverse axis \( a \) is:
\[ a = \frac{3}{2} \]

### Final Answer

The length of the transverse axis is:
\[ \text{Length of the transverse axis} = 3 \]