Two non-intersecting circles, one lying inside the other arc of radius x and `y(x gt y)`. If the minimum distance between circumference is z. Then, the distance between their centres is

Options

A. `x + z - y`
B. `x - z - y`
C. `x - z + y`
D. None of these

Answer

To find the distance between the centers of two non-intersecting circles, one lying inside the other, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Let the radius of the larger circle (outer circle) be \( x \).
- Let the radius of the smaller circle (inner circle) be \( y \).
- The minimum distance between the circumferences of the two circles is given as \( z \).

2. **Visualize the Circles**:
- Draw two circles, one inside the other. Label the center of the larger circle as \( O_1 \) and the center of the smaller circle as \( O_2 \).
- The radius of the larger circle is \( O_1A = x \) and the radius of the smaller circle is \( O_2B = y \).

3. **Understand the Relationship**:
- The distance from the center of the larger circle to the circumference is \( x \).
- The distance from the center of the smaller circle to the circumference is \( y \).
- The minimum distance between the circumferences of the two circles is \( z \).

4. **Set Up the Equation**:
- The distance between the centers \( O_1O_2 \) can be expressed as:
\[ O_1O_2 = O_1A - O_2B + z \]
- Here, \( O_1A \) is the radius of the larger circle, \( O_2B \) is the radius of the smaller circle, and \( z \) is the distance between the circumferences.

5. **Substitute the Values**:
- Substitute \( O_1A = x \) and \( O_2B = y \) into the equation:
\[ O_1O_2 = x - y + z \]

6. **Final Expression**:
- Therefore, the distance between the centers of the two circles is:
\[ O_1O_2 = x - y + z \]

### Conclusion:
The distance between the centers of the two circles is \( x - y + z \).

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