The length of a common internal tangent to two circles is 5 and that of a common external tangent is 13. If the product of the radii of two circles is `lamda`, then the value of `(lamda)/(4)` is

Answer

To solve the problem, we will use the formulas for the lengths of the common internal and external tangents between two circles. Let's denote the radii of the two circles as \( r_1 \) and \( r_2 \).

### Step-by-Step Solution:

1. **Identify the lengths of the tangents**:
- The length of the common internal tangent is given as \( L_i = 5 \).
- The length of the common external tangent is given as \( L_e = 13 \).

2. **Use the formulas for the lengths of the tangents**:
- The formula for the length of the common internal tangent is:
\[ L_i = \sqrt{d^2 - (r_1 + r_2)^2} \]
- The formula for the length of the common external tangent is:
\[ L_e = \sqrt{d^2 - (r_1 - r_2)^2} \]
Where \( d \) is the distance between the centers of the two circles.

3. **Set up the equations**:
- From the internal tangent:
\[ 5 = \sqrt{d^2 - (r_1 + r_2)^2} \]
Squaring both sides gives:
\[ 25 = d^2 - (r_1 + r_2)^2 \quad \text{(1)} \]
- From the external tangent:
\[ 13 = \sqrt{d^2 - (r_1 - r_2)^2} \]
Squaring both sides gives:
\[ 169 = d^2 - (r_1 - r_2)^2 \quad \text{(2)} \]

4. **Subtract the two equations**:
- Subtract equation (1) from equation (2):
\[ 169 - 25 = (d^2 - (r_1 - r_2)^2) - (d^2 - (r_1 + r_2)^2) \]
This simplifies to:
\[ 144 = (r_1 + r_2)^2 - (r_1 - r_2)^2 \]

5. **Use the difference of squares**:
- The difference of squares can be factored:
\[ (a^2 - b^2) = (a - b)(a + b) \]
- Let \( a = r_1 + r_2 \) and \( b = r_1 - r_2 \):
\[ 144 = (r_1 + r_2 - (r_1 - r_2))(r_1 + r_2 + (r_1 - r_2)) \]
This simplifies to:
\[ 144 = (2r_2)(2r_1) = 4r_1r_2 \]

6. **Solve for the product of the radii**:
- Rearranging gives:
\[ r_1r_2 = \frac{144}{4} = 36 \]
- Therefore, \( \lambda = r_1r_2 = 36 \).

7. **Find \( \frac{\lambda}{4} \)**:
- Finally, we calculate:
\[ \frac{\lambda}{4} = \frac{36}{4} = 9 \]

### Final Answer:
\[ \frac{\lambda}{4} = 9 \]