The distance between the centres of the two circles of radii,`r_1` and `r_2` is d. They will touch each other internally if

Options

A. `d = r_1 or r_2`
B. `d = r_1 + r_2`
C. `d = r_1 - r_2`
D. `d = sqrt(r_1 r_2)`

Answer

To determine the condition under which two circles of radii \( r_1 \) and \( r_2 \) touch each other internally, we need to analyze the relationship between the distance \( d \) between their centers and their radii.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have two circles: one with radius \( r_1 \) and the other with radius \( r_2 \).
- Let the centers of these circles be \( O_1 \) and \( O_2 \) respectively.
- The distance between the centers \( O_1 \) and \( O_2 \) is given as \( d \).

2. **Condition for Internal Touching**:
- For the circles to touch each other internally, the distance \( d \) between the centers must be equal to the difference of their radii.
- This means that the larger circle (with radius \( r_2 \)) encompasses the smaller circle (with radius \( r_1 \)).

3. **Mathematical Representation**:
- The condition for internal touching can be expressed as:
\[ d = r_2 - r_1 \]
- Here, \( r_2 \) is the radius of the larger circle, and \( r_1 \) is the radius of the smaller circle.

4. **Conclusion**:
- Thus, the circles will touch each other internally if the distance \( d \) between their centers is equal to the difference of their radii.

### Final Condition:
- The circles will touch each other internally if:
\[ d = r_2 - r_1 \]