The number of common tangents of the circles `x^(2)+y^(2)+4x+1=0` and `x^(2)+y^(2)-2y-7=0`, is

Options

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

Answer

To find the number of common tangents of the circles given by the equations \(x^2 + y^2 + 4x + 1 = 0\) and \(x^2 + y^2 - 2y - 7 = 0\), we will follow these steps:

### Step 1: Rewrite the equations of the circles in standard form

1. **Circle 1**: \(x^2 + y^2 + 4x + 1 = 0\)
- Rearranging gives:
\[ (x^2 + 4x) + y^2 + 1 = 0 \]
- Completing the square for \(x\):
\[ (x + 2)^2 - 4 + y^2 + 1 = 0 \implies (x + 2)^2 + y^2 = 3 \]
- This means Circle 1 has center \((-2, 0)\) and radius \(r_1 = \sqrt{3}\).

2. **Circle 2**: \(x^2 + y^2 - 2y - 7 = 0\)
- Rearranging gives:
\[ x^2 + (y^2 - 2y) - 7 = 0 \]
- Completing the square for \(y\):
\[ x^2 + (y - 1)^2 - 1 - 7 = 0 \implies x^2 + (y - 1)^2 = 8 \]
- This means Circle 2 has center \((0, 1)\) and radius \(r_2 = \sqrt{8} = 2\sqrt{2}\).

### Step 2: Find the distance between the centers of the circles

- The distance \(d\) between the centers \((-2, 0)\) and \((0, 1)\) is calculated as:
\[ d = \sqrt{(-2 - 0)^2 + (0 - 1)^2} = \sqrt{4 + 1} = \sqrt{5} \]

### Step 3: Determine the conditions for the number of common tangents

- We need to compare the distance \(d\) with the sum and difference of the radii:
- Sum of the radii: \(r_1 + r_2 = \sqrt{3} + 2\sqrt{2}\)
- Difference of the radii: \(|r_1 - r_2| = |\sqrt{3} - 2\sqrt{2}|\)

### Step 4: Calculate the values

1. **Calculate \(r_1 + r_2\)**:
- Approximating:
\[ \sqrt{3} \approx 1.732, \quad 2\sqrt{2} \approx 2.828 \implies r_1 + r_2 \approx 1.732 + 2.828 \approx 4.560 \]

2. **Calculate \(|r_1 - r_2|\)**:
- Approximating:
\[ |\sqrt{3} - 2\sqrt{2}| \approx |1.732 - 2.828| \approx | -1.096 | \approx 1.096 \]

### Step 5: Analyze the conditions

- Now we have:
- \(d = \sqrt{5} \approx 2.236\)
- \(r_1 + r_2 \approx 4.560\)
- \(|r_1 - r_2| \approx 1.096\)

- The conditions for the number of common tangents are:
- If \(d > r_1 + r_2\): No common tangents
- If \(d = r_1 + r_2\): One common tangent
- If \(|r_1 - r_2| < d < r_1 + r_2\): Two common tangents
- If \(d = |r_1 - r_2|\): One common tangent (internally)
- If \(d < |r_1 - r_2|\): No common tangents (one circle inside the other)

### Conclusion

Since \(1.096 < 2.236 < 4.560\), we conclude that there are **2 common tangents**.

### Final Answer:
The number of common tangents of the circles is **2**.