R and r are the radii of two circles (R > r). If the distance between the centres of the two circles be d, then length of common tangent of two circles is

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( R \) be the radius of the larger circle.
- Let \( r \) be the radius of the smaller circle.
- Let \( d \) be the distance between the centers of the two circles.

2. **Understand the Geometry**:
- We have two circles: Circle 1 with center \( O_1 \) and radius \( R \), and Circle 2 with center \( O_2 \) and radius \( r \).
- The distance between the centers \( O_1 \) and \( O_2 \) is \( d \).

3. **Apply the Formula for the Length of the Common Tangent**:
- The formula for the length of the common tangent \( L \) between two circles is given by:
\[ L = \sqrt{d^2 - (R - r)^2} \]
- This formula derives from the Pythagorean theorem applied to the right triangle formed by the tangent line and the radii to the points of tangency.

4. **Substitute the Values**:
- Substitute the known values into the formula:
\[ L = \sqrt{d^2 - (R - r)^2} \]

5. **Simplify the Expression**:
- Calculate \( (R - r)^2 \) and then substitute it back into the equation to find the length of the common tangent.

### Final Result:
The length of the common tangent between the two circles is:
\[ L = \sqrt{d^2 - (R - r)^2} \]