Distance between centre of two circles is 4.5 cm, their radius are 2 cm and 2.5 cm, respectively, the number of tangents that can be drawn on them are equal to

Options

A. 1
B. 2
C. 3
D. None of these

Answer

To determine the number of common tangents that can be drawn to two circles, we can use the following steps:

### Step 1: Identify the given information
- Distance between the centers of the circles (d) = 4.5 cm
- Radius of the first circle (r1) = 2 cm
- Radius of the second circle (r2) = 2.5 cm

### Step 2: Determine the relationship between the distance and the radii
To find the number of common tangents, we need to check the relationship between the distance between the centers (d) and the sum and difference of the radii (r1 and r2).

- Calculate the sum of the radii:
\[ r1 + r2 = 2 + 2.5 = 4.5 \text{ cm} \]

- Calculate the absolute difference of the radii:
\[ |r1 - r2| = |2 - 2.5| = 0.5 \text{ cm} \]

### Step 3: Analyze the conditions for common tangents
The number of common tangents can be determined using the following conditions:
1. If \( d > r1 + r2 \): 2 external tangents
2. If \( d = r1 + r2 \): 1 external tangent (the circles touch externally)
3. If \( |r1 - r2| < d < r1 + r2 \): 2 external and 2 internal tangents
4. If \( d = |r1 - r2| \): 1 internal tangent (the circles touch internally)
5. If \( d < |r1 - r2| \): No tangents

### Step 4: Apply the conditions
In our case:
- \( d = 4.5 \text{ cm} \)
- \( r1 + r2 = 4.5 \text{ cm} \)

Since \( d = r1 + r2 \), it means the circles are touching each other externally.

### Step 5: Conclusion
Since the circles touch externally, there is exactly **1 common tangent** that can be drawn.

### Final Answer:
The number of tangents that can be drawn on them is **1**.

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