The distance between the centre of two circles having radii 5 cm and 6 cm is `sqrt(10)` cm. find the length (in cm) of direct common tangent.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the length of the direct common tangent between two circles with given radii and distance between their centers, 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) = 5 cm
- Radius of the second circle (r2) = 6 cm
- Distance between the centers of the circles (d) = √10 cm

2. **Calculate the difference of the radii:**
- Difference of the radii (|r1 - r2|) = |5 - 6| = 1 cm

3. **Use the formula for the length of the direct common tangent:**
The formula for the length (L) of the direct common tangent between two circles is given by:
\[ L = \sqrt{d^2 - (r1 - r2)^2} \]
where:
- d = distance between the centers of the circles
- r1 = radius of the first circle
- r2 = radius of the second circle

4. **Substitute the values into the formula:**
\[ L = \sqrt{(\sqrt{10})^2 - (1)^2} \]
\[ L = \sqrt{10 - 1} \]
\[ L = \sqrt{9} \]

5. **Calculate the length of the direct common tangent:**
\[ L = 3 \text{ cm} \]

### Final Answer:
The length of the direct common tangent is **3 cm**.

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