Circles with radii 3, 4 and 5 touch each other externally. If P is the point of intersection of tangents to these circles at their points of contact, then if the distance of P from the points of contact is `lambda` then `lambda_2` is _____

Explanation

To solve the problem, we need to find the value of \( \lambda_2 \), which is related to the distance of point \( P \) from the points of contact of the circles with radii 3, 4, and 5 that touch each other externally.

### Step-by-Step Solution:

1. **Identify the Radii of the Circles**:
- Let the radii of the circles be \( r_1 = 3 \), \( r_2 = 4 \), and \( r_3 = 5 \).

2. **Calculate the Semi-Perimeter \( s \)**:
- The distances between the centers of the circles are:
- Distance between Circle 1 and Circle 2: \( r_1 + r_2 = 3 + 4 = 7 \)
- Distance between Circle 2 and Circle 3: \( r_2 + r_3 = 4 + 5 = 9 \)
- Distance between Circle 3 and Circle 1: \( r_3 + r_1 = 5 + 3 = 8 \)
- The semi-perimeter \( s \) is calculated as:
\[ s = \frac{(r_1 + r_2) + (r_2 + r_3) + (r_3 + r_1)}{2} = \frac{7 + 9 + 8}{2} = \frac{24}{2} = 12 \]

3. **Calculate the Area \( \Delta \)**:
- Using Heron's formula, the area \( \Delta \) of the triangle formed by the centers of the circles is:
\[ \Delta = \sqrt{s(s - a)(s - b)(s - c)} \]
where \( a = 7 \), \( b = 9 \), and \( c = 8 \).
- Thus, we compute:
\[ \Delta = \sqrt{12(12 - 7)(12 - 9)(12 - 8)} = \sqrt{12 \cdot 5 \cdot 3 \cdot 4} \]
\[ = \sqrt{720} = 12\sqrt{5} \]

4. **Calculate the Inner Radius \( r \)**:
- The inner radius \( r \) is given by:
\[ r = \frac{\Delta}{s} = \frac{12\sqrt{5}}{12} = \sqrt{5} \]

5. **Determine \( \lambda \) and \( \lambda_2 \)**:
- The distance from point \( P \) to the points of contact is given by \( \lambda \), which is equal to the inner radius \( r \).
- Therefore, \( \lambda = \sqrt{5} \).
- To find \( \lambda_2 \), we square \( \lambda \):
\[ \lambda_2 = \lambda^2 = (\sqrt{5})^2 = 5 \]

### Final Answer:
\[ \lambda_2 = 5 \]