R and r are the radius of two circles `(R gt r)`. If the distance between the centre of the two circles be `d`, then length of common tangent of two circles is

Options

A. `sqrt( d^(2) - (R- r) ^2)`
B. `sqrt( ( R - r)^(2) - d^(2) )`
C. `sqrt( R^(2) - d^(2) )`
D. `sqrt( r^(2) - d^(2) )`

Answer

To find the length of the common tangent of two circles with radii \( R \) and \( r \) (where \( R > r \)) and a distance \( d \) between their centers, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have two circles: Circle 1 with radius \( R \) and Circle 2 with radius \( r \).
- The distance between the centers of these circles is \( d \).

2. **Identify the Points**:
- Let \( A \) be the center of the larger circle (radius \( R \)).
- Let \( B \) be the center of the smaller circle (radius \( r \)).
- The points where the common tangent touches Circle 1 and Circle 2 can be denoted as \( P \) and \( Q \), respectively.

3. **Use the Tangent Length Formula**:
- The length of the common tangent \( AB \) can be calculated using the formula:
\[ AB = \sqrt{d^2 - (R - r)^2} \]
- Here, \( d \) is the distance between the centers, and \( R - r \) is the difference in the radii of the two circles.

4. **Substitute the Values**:
- Substitute the values of \( d \), \( R \), and \( r \) into the formula to find the length of the common tangent.

5. **Final Expression**:
- The final expression for the length of the common tangent is:
\[ AB = \sqrt{d^2 - (R - r)^2} \]