A circle of radius 2 units is touching both the axes and a circle with centre at (6,5). The distance between their centres is

Options

A. 8 units
B. 5 units
C. 7 units
D. none of these

Answer

To solve the problem, we need to find the distance between the centers of two circles. One circle has a radius of 2 units and is touching both the x-axis and y-axis, while the other circle has its center at (6, 5).

### Step-by-Step Solution:

1. **Identify the center of the first circle:**
- The first circle has a radius of 2 units and is touching both the x-axis and y-axis. Therefore, its center will be at (2, 2).
- **Hint:** The center of a circle that touches the axes at a distance equal to its radius can be found by adding the radius to the origin (0,0).

2. **Identify the center of the second circle:**
- The second circle has its center given as (6, 5).
- **Hint:** The coordinates of the center of the second circle are provided directly in the problem.

3. **Calculate the distance between the two centers:**
- We will use the distance formula to find the distance between the points (2, 2) and (6, 5).
- The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Here, \((x_1, y_1) = (2, 2)\) and \((x_2, y_2) = (6, 5)\).
- Substituting the values:
\[ d = \sqrt{(6 - 2)^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} \]
- Simplifying further:
\[ d = \sqrt{16 + 9} = \sqrt{25} = 5 \]

4. **Conclusion:**
- The distance between the centers of the two circles is 5 units.

### Final Answer:
The distance between the centers of the circles is **5 units**.