The curvatures of two externally tangent circles are 0.5 and 2, what is the distance between their centers

Options

A. A. 2.5
B. B. 1
C. C. 4
D. D. 8

Answer

To find the distance between the centers of two externally tangent circles given their curvatures, we can follow these steps:

### Step 1: Understand the concept of curvature
Curvature (k) of a circle is defined as the reciprocal of its radius (r). Therefore, if we know the curvature, we can find the radius using the formula:
\[ r = \frac{1}{k} \]

### Step 2: Calculate the radii of the circles
Given the curvatures:
- For the first circle, \( k_1 = 0.5 \)
- For the second circle, \( k_2 = 2 \)

We can calculate the radii:
- Radius of the first circle \( r_1 = \frac{1}{k_1} = \frac{1}{0.5} = 2 \)
- Radius of the second circle \( r_2 = \frac{1}{k_2} = \frac{1}{2} = 0.5 \)

### Step 3: Use the formula for the distance between the centers
The distance \( d \) between the centers of two externally tangent circles is given by the sum of their radii:
\[ d = r_1 + r_2 \]

### Step 4: Substitute the values
Now, substitute the values of \( r_1 \) and \( r_2 \):
\[ d = 2 + 0.5 = 2.5 \]

### Step 5: Conclusion
Thus, the distance between the centers of the two circles is \( 2.5 \).

### Final Answer
The distance between the centers of the two circles is \( 2.5 \).

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