The distance between the centres of the two circles of radii `r_1` and `r_2` is d. They will thouch 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 with radii \( r_1 \) and \( r_2 \) touch each other internally, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Internal Tangency**: When two circles touch each other internally, one circle is inside the other, and they meet at exactly one point.

2. **Identifying the Centers and Radii**: Let the radius of the larger circle be \( r_1 \) and the radius of the smaller circle be \( r_2 \). The distance between the centers of the two circles is denoted as \( d \).

3. **Setting Up the Relationship**: For the circles to touch internally, the distance \( d \) between their centers must equal the difference in their radii. This is because the distance from the center of the larger circle to the point of tangency is \( r_1 \), and the distance from the center of the smaller circle to the same point is \( r_2 \).

4. **Formulating the Condition**: Therefore, the condition for the circles to touch each other internally can be expressed mathematically as:
\[ d = r_1 - r_2 \]

5. **Conclusion**: Thus, the circles will touch each other internally if the distance \( d \) between their centers is equal to the difference of their radii \( r_1 \) and \( r_2 \).

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