If diameters of two circles are 6 units and 10 units and their centres are 8 unit apart, how many tangents line can be drawn to the circle?

Options

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

Answer

To determine how many tangent lines can be drawn to two circles given their diameters and the distance between their centers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the diameters and calculate the radii of the circles:**
- Diameter of Circle 1 (D1) = 6 units
- Diameter of Circle 2 (D2) = 10 units
- Radius of Circle 1 (R1) = D1 / 2 = 6 / 2 = 3 units
- Radius of Circle 2 (R2) = D2 / 2 = 10 / 2 = 5 units

2. **Determine the distance between the centers of the circles:**
- Distance (d) between the centers = 8 units

3. **Apply the conditions for the number of tangents:**
- We have the following conditions to determine the number of tangents:
- If \( d > R1 + R2 \): 4 tangents
- If \( d = R1 + R2 \): 3 tangents
- If \( |R1 - R2| < d < R1 + R2 \): 2 tangents
- If \( d = |R1 - R2| \): 1 tangent
- If \( d < |R1 - R2| \): 0 tangents

4. **Calculate \( R1 + R2 \) and \( |R1 - R2| \):**
- \( R1 + R2 = 3 + 5 = 8 \)
- \( |R1 - R2| = |3 - 5| = 2 \)

5. **Compare the distance with the calculated values:**
- We have \( d = 8 \) which is equal to \( R1 + R2 \).
- Therefore, according to the conditions, since \( d = R1 + R2 \), there are 3 tangents.

6. **Conclusion:**
- The number of tangent lines that can be drawn to the circles is **3**.