The length of a common internal tangent to two circles is 5 and a common external tangent is 15, then the product of the radii of the two circles is :

Options

A. 25
B. 50
C. 75
D. 30

Answer

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

Given:
- Length of the common internal tangent \( m = 5 \)
- Length of the common external tangent \( M = 15 \)

### Step 1: Write the formulas for the lengths of the tangents.

The formulas for the lengths of the common tangents are:
1. For the internal tangent:
\[ m^2 = d^2 - (r_1 + r_2)^2 \]
2. For the external tangent:
\[ M^2 = d^2 - (r_1 - r_2)^2 \]

### Step 2: Substitute the known values into the formulas.

Substituting \( m = 5 \) and \( M = 15 \):
1. For the internal tangent:
\[ 5^2 = d^2 - (r_1 + r_2)^2 \implies 25 = d^2 - (r_1 + r_2)^2 \quad \text{(1)} \]
2. For the external tangent:
\[ 15^2 = d^2 - (r_1 - r_2)^2 \implies 225 = d^2 - (r_1 - r_2)^2 \quad \text{(2)} \]

### Step 3: Rearrange both equations to express \( d^2 \).

From equation (1):
\[ d^2 = 25 + (r_1 + r_2)^2 \quad \text{(3)} \]

From equation (2):
\[ d^2 = 225 + (r_1 - r_2)^2 \quad \text{(4)} \]

### Step 4: Set equations (3) and (4) equal to each other.

Since both equations equal \( d^2 \), we can set them equal:
\[ 25 + (r_1 + r_2)^2 = 225 + (r_1 - r_2)^2 \]

### Step 5: Simplify the equation.

Rearranging gives:
\[ (r_1 + r_2)^2 - (r_1 - r_2)^2 = 225 - 25 \]
\[ (r_1 + r_2)^2 - (r_1 - r_2)^2 = 200 \]

### Step 6: Use the difference of squares.

Using the identity \( a^2 - b^2 = (a-b)(a+b) \):
Let \( a = r_1 + r_2 \) and \( b = r_1 - r_2 \):
\[ [(r_1 + r_2) - (r_1 - r_2)][(r_1 + r_2) + (r_1 - r_2)] = 200 \]
This simplifies to:
\[ (2r_2)(2r_1) = 200 \]
\[ 4r_1 r_2 = 200 \]

### Step 7: Solve for the product of the radii.

Dividing both sides by 4:
\[ r_1 r_2 = 50 \]

### Final Answer:
The product of the radii of the two circles is \( \boxed{50} \).