The distance between the centres of two circles having radii 5 cm and 6 cm is 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 the distance between their centers, we can use the following steps:

### Step 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

### Step 2: 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:
\[ L = \sqrt{d^2 - (r1 + r2)^2} \]

### Step 3: Calculate the sum of the radii
Calculate \( r1 + r2 \):
\[ r1 + r2 = 5 + 6 = 11 \text{ cm} \]

### Step 4: Substitute the values into the formula
Now substitute the values into the formula:
\[ L = \sqrt{10^2 - 11^2} \]
\[ L = \sqrt{100 - 121} \]
\[ L = \sqrt{-21} \]

### Step 5: Interpret the result
Since the result under the square root is negative, it indicates that there is no direct common tangent between the two circles. This occurs because the sum of the radii (11 cm) is greater than the distance between the centers (10 cm).

### Final Answer
The length of the direct common tangent does not exist in this case.

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