The length of the direct common tangent of two circles of radii 2 cm and 8 cm is 8 cm. Then the distance between the centres of the circles is

Options

A. 8 cm
B. 10 cm
C. 6 cm
D. 12 cm

Answer

To find the distance between the centers of two circles with given radii and the length of the direct common tangent, we can use the formula for the length of the direct common tangent.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the first circle (R1) = 2 cm
- Radius of the second circle (R2) = 8 cm
- Length of the direct common tangent (L) = 8 cm

2. **Use the Formula for the Length of the Direct Common Tangent:**
The formula for the length of the direct common tangent (L) between two circles is given by:
\[ L = \sqrt{d^2 - (R1 + R2)^2} \]
where \(d\) is the distance between the centers of the two circles.

3. **Substitute the Known Values into the Formula:**
We can rearrange the formula to find \(d\):
\[ L^2 = d^2 - (R1 + R2)^2 \]
Substituting the known values:
\[ 8^2 = d^2 - (2 + 8)^2 \]
Simplifying this gives:
\[ 64 = d^2 - 100 \]

4. **Solve for \(d^2\):**
Rearranging the equation:
\[ d^2 = 64 + 100 \]
\[ d^2 = 164 \]

5. **Calculate \(d\):**
Taking the square root of both sides:
\[ d = \sqrt{164} \]
Simplifying further:
\[ d = \sqrt{4 \times 41} = 2\sqrt{41} \]

6. **Approximate the Value of \(d\):**
To get a numerical value, we can calculate:
\[ \sqrt{41} \approx 6.4 \quad \text{(approximately)} \]
Therefore:
\[ d \approx 2 \times 6.4 = 12.8 \text{ cm} \]

### Final Answer:
The distance between the centers of the circles is approximately \(12.8\) cm.