To find the number of possible common tangents of the given pairs of circles, we will follow these steps:
### Step 1: Convert the circle equations to standard form
The general equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.
### Step 2: Identify the centers and radii
From the given equations, we will extract the centers and radii for each pair of circles.
### Step 3: Calculate the distance between the centers
The distance \(d\) between the centers of two circles with centers \((h_1, k_1)\) and \((h_2, k_2)\) is given by:
\[
d = \sqrt{(h_2 - h_1)^2 + (k_2 - k_1)^2}
\]
### Step 4: Determine the number of common tangents
Using the following conditions:
1. If \(d > r_1 + r_2\), there are **4 common tangents** (2 direct and 2 transverse).
2. If \(d = r_1 + r_2\), there are **3 common tangents** (1 at the point of contact and 2 direct).
3. If \(|r_1 - r_2| < d < r_1 + r_2\), there are **2 common tangents** (2 direct).
4. If \(d = |r_1 - r_2|\), there is **1 common tangent** (1 at the point of contact).
5. If \(d < |r_1 - r_2|\), there are **0 common tangents**.
### Step 5: Apply the above steps to each pair of circles
#### (i) Circles:
\[
x^2 + y^2 - 14x + 6y + 33 = 0 \quad \text{and} \quad x^2 + y^2 + 30x - 2y + 1 = 0
\]
1. **Convert to standard form**:
- Circle 1: Center \((7, -3)\), Radius \(5\)
- Circle 2: Center \((-15, 1)\), Radius \(15\)
2. **Distance between centers**:
\[
d = \sqrt{(-15 - 7)^2 + (1 + 3)^2} = \sqrt{(-22)^2 + (4)^2} = \sqrt{484 + 16} = \sqrt{500} = 10\sqrt{5}
\]
3. **Check conditions**:
\[
10\sqrt{5} \approx 22.36 > 5 + 15 = 20 \quad \text{(4 common tangents)}
\]
#### (ii) Circles:
\[
x^2 + y^2 + 6x + 6y + 14 = 0 \quad \text{and} \quad x^2 + y^2 - 2x - 4y - 4 = 0
\]
1. **Convert to standard form**:
- Circle 1: Center \((-3, -3)\), Radius \(2\)
- Circle 2: Center \((1, 2)\), Radius \(3\)
2. **Distance between centers**:
\[
d = \sqrt{(1 + 3)^2 + (2 + 3)^2} = \sqrt{(4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41}
\]
3. **Check conditions**:
\[
\sqrt{41} \approx 6.4 > 2 + 3 = 5 \quad \text{(4 common tangents)}
\]
#### (iii) Circles:
\[
x^2 + y^2 - 4x - 2y + 1 = 0 \quad \text{and} \quad x^2 + y^2 - 6x - 4y + 4 = 0
\]
1. **Convert to standard form**:
- Circle 1: Center \((2, 1)\), Radius \(1\)
- Circle 2: Center \((3, 2)\), Radius \(3\)
2. **Distance between centers**:
\[
d = \sqrt{(3 - 2)^2 + (2 - 1)^2} = \sqrt{(1)^2 + (1)^2} = \sqrt{2}
\]
3. **Check conditions**:
\[
\sqrt{2} < 3 - 1 = 2 \quad \text{(0 common tangents)}
\]
#### (iv) Circles:
\[
x^2 + y^2 - 4x + 2y - 4 = 0 \quad \text{and} \quad x^2 + y^2 + 2x - 6y + 6 = 0
\]
1. **Convert to standard form**:
- Circle 1: Center \((2, -1)\), Radius \(3\)
- Circle 2: Center \((-1, 3)\), Radius \(2\)
2. **Distance between centers**:
\[
d = \sqrt{(-1 - 2)^2 + (3 + 1)^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = 5
\]
3. **Check conditions**:
\[
5 = 3 + 2 \quad \text{(3 common tangents)}
\]
#### (v) Circles:
\[
x^2 + y^2 + 4x - 6y - 3 = 0 \quad \text{and} \quad x^2 + y^2 + 4x - 2y + 4 = 0
\]
1. **Convert to standard form**:
- Circle 1: Center \((-2, 3)\), Radius \(4\)
- Circle 2: Center \((-2, 1)\), Radius \(1\)
2. **Distance between centers**:
\[
d = \sqrt{(-2 - (-2))^2 + (1 - 3)^2} = \sqrt{0 + (-2)^2} = 2
\]
3. **Check conditions**:
\[
2 < 4 - 1 = 3 \quad \text{(0 common tangents)}
\]
### Final Results
1. **(i)** 4 common tangents
2. **(ii)** 4 common tangents
3. **(iii)** 0 common tangents
4. **(iv)** 3 common tangents
5. **(v)** 0 common tangents
