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This question involves the concept of tangents to two circles, specifically the length of common external and internal tangents. Here's how we can approach the problem:

The formulas for the lengths of the common external and internal tangents between two circles with radii \( r_1 \) and \( r_2 \) and distance between their centers \( d \) are:

- **External Tangent Length**: 
  \[
  L_{\text{ext}} = \sqrt{d^2 - (r_1 - r_2)^2}
  \]
  
- **Internal Tangent Length**: 
  \[
  L_{\text{int}} = \sqrt{d^2 - (r_1 + r_2)^2}
  \]

Given:
- Length of the external tangent: \( L_{\text{ext}} = 9 \)
- Length of the internal tangent: \( L_{\text{int}} = 5 \)

We need to find the product of the radii, \( r_1 \times r_2 \).

To proceed, let's use the given data and solve for \( r_1 \times r_2 \).

1. **Using the formulas for external and internal tangents**:
   \[
   L_{\text{ext}} = \sqrt{d^2 - (r_1 - r_2)^2} = 9
   \]
   \[
   L_{\text{int}} = \sqrt{d^2 - (r_1 + r_2)^2} = 5
   \]

2. **Square both equations**:
   \[
   d^2 - (r_1 - r_2)^2 = 81 \quad \text{(from external tangent)}
   \]
   \[
   d^2 - (r_1 + r_2)^2 = 25 \quad \text{(from internal tangent)}
   \]

3. **Subtract the second equation from the first**:
   \[
   \left[d^2 - (r_1 - r_2)^2\right] - \left[d^2 - (r_1 + r_2)^2\right] = 81 - 25
   \]
   Simplifying:
   \[
   (r_1 + r_2)^2 - (r_1 - r_2)^2 = 56
   \]
   Using the identity \( (a^2 - b^2) = (a - b)(a + b) \), this becomes:
   \[
   4r_1r_2 = 56
   \]
   Therefore:
   \[
   r_1r_2 = 14
   \]

Thus, the product of the radii is **14**, which corresponds to option C.

Would you like more details on the process, or have any questions?

Here are some related questions to consider:
1. How do we derive the formulas for internal and external tangents between circles?
2. What is the significance of the distance between the centers in these tangent formulas?
3. Can we solve this problem if only the external tangent length is given?
4. How do internal and external tangents differ geometrically?
5. What happens if the circles overlap or are concentric in such problems?

**Tip:** Remember that subtraction of the squared forms often simplifies problems involving geometric properties like distances and tangents!

If the length of a common external tangent to two circles is 9 and that of a common internal tangent is 5, then the product of the radii of the two circles is:

Solve the geometry problem where the length of the common external tangent is 9 and the internal tangent is 5. Learn how to calculate the product of the radii of two circles using tangent properties and algebraic techniques. Ideal for students in Grades 10-12.

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