Number of common tangents to the circles
`C_(1):x ^(2) + y ^(2) - 6x - 4y -12=0`
and ` C _(2) : x ^(2) + y ^(2) + 6x + 4y+ 4=0` is

Answer

To find the number of common tangents to the circles given by the equations:

1. \( C_1: x^2 + y^2 - 6x - 4y - 12 = 0 \)
2. \( C_2: x^2 + y^2 + 6x + 4y + 4 = 0 \)

we will follow these steps:

### Step 1: Rewrite the equations in standard form
We start by rewriting each circle's equation in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \).

**For Circle \( C_1 \):**
\[ x^2 + y^2 - 6x - 4y - 12 = 0 \]
Rearranging gives:
\[ x^2 - 6x + y^2 - 4y = 12 \]
Completing the square for \( x \) and \( y \):
\[ (x - 3)^2 - 9 + (y - 2)^2 - 4 = 12 \]
\[ (x - 3)^2 + (y - 2)^2 = 25 \]
Thus, the center \( C_1 \) is \( (3, 2) \) and the radius \( r_1 = 5 \).

**For Circle \( C_2 \):**
\[ x^2 + y^2 + 6x + 4y + 4 = 0 \]
Rearranging gives:
\[ x^2 + 6x + y^2 + 4y = -4 \]
Completing the square for \( x \) and \( y \):
\[ (x + 3)^2 - 9 + (y + 2)^2 - 4 = -4 \]
\[ (x + 3)^2 + (y + 2)^2 = 9 \]
Thus, the center \( C_2 \) is \( (-3, -2) \) and the radius \( r_2 = 3 \).

### Step 2: Calculate the distance between the centers
Now we calculate the distance \( d \) between the centers \( C_1(3, 2) \) and \( C_2(-3, -2) \):
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{((-3) - 3)^2 + ((-2) - 2)^2} \]
\[ = \sqrt{(-6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]

### Step 3: Determine the number of common tangents
To determine the number of common tangents, we compare the distance \( d \) with the sum and difference of the radii \( r_1 \) and \( r_2 \):

- \( r_1 + r_2 = 5 + 3 = 8 \)
- \( |r_1 - r_2| = |5 - 3| = 2 \)

Now we check the conditions:
1. If \( d > r_1 + r_2 \): 2 external tangents
2. If \( d = r_1 + r_2 \): 1 external tangent
3. If \( |r_1 - r_2| < d < r_1 + r_2 \): 2 external and 2 internal tangents
4. If \( d = |r_1 - r_2| \): 1 internal tangent
5. If \( d < |r_1 - r_2| \): No common tangents

Here, we have:
- \( d = 2\sqrt{13} \approx 7.21 \)
- \( r_1 + r_2 = 8 \)
- \( |r_1 - r_2| = 2 \)

Since \( 2 < d < 8 \), we have 2 external tangents.

### Final Answer
The number of common tangents to the circles \( C_1 \) and \( C_2 \) is **2**.

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