Find the number of common tangents of the circles `x^2+y^2-2x-6y+9=0` and `x^2+y^2+6x-2y+1=0`

Options

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

Explanation

To find the number of common tangents of the circles given by the equations \(x^2 + y^2 - 2x - 6y + 9 = 0\) and \(x^2 + y^2 + 6x - 2y + 1 = 0\), we can follow these steps:

### Step 1: Write the equations in standard form
We start by rewriting the given equations in the standard form of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\).

1. **First Circle:**
\[ x^2 + y^2 - 2x - 6y + 9 = 0 \]
Rearranging gives:
\[ (x^2 - 2x) + (y^2 - 6y) = -9 \]
Completing the square:
\[ (x - 1)^2 - 1 + (y - 3)^2 - 9 = -9 \]
\[ (x - 1)^2 + (y - 3)^2 = 1 \]
This gives us the center \(C_1(1, 3)\) and radius \(r_1 = 1\).

2. **Second Circle:**
\[ x^2 + y^2 + 6x - 2y + 1 = 0 \]
Rearranging gives:
\[ (x^2 + 6x) + (y^2 - 2y) = -1 \]
Completing the square:
\[ (x + 3)^2 - 9 + (y - 1)^2 - 1 = -1 \]
\[ (x + 3)^2 + (y - 1)^2 = 9 \]
This gives us the center \(C_2(-3, 1)\) and radius \(r_2 = 3\).

### Step 2: Calculate the distance between the centers
Next, we find the distance \(d\) between the centers \(C_1\) and \(C_2\):
\[ d = \sqrt{(1 - (-3))^2 + (3 - 1)^2} = \sqrt{(1 + 3)^2 + (3 - 1)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]

### Step 3: Determine the number of common tangents
To find the number of common tangents, we compare \(d\) with \(r_1 + r_2\) and \(|r_1 - r_2|\):

1. Calculate \(r_1 + r_2\):
\[ r_1 + r_2 = 1 + 3 = 4 \]

2. Calculate \(|r_1 - r_2|\):
\[ |r_1 - r_2| = |1 - 3| = 2 \]

Now, we check the conditions:
- If \(d > r_1 + r_2\), there are 4 common tangents (2 external and 2 internal).
- If \(d = r_1 + r_2\), there are 3 common tangents (2 external and 1 internal).
- If \(|r_1 - r_2| < d < r_1 + r_2\), there are 2 common tangents (both external).
- If \(d = |r_1 - r_2|\), there is 1 common tangent (internal).
- If \(d < |r_1 - r_2|\), there are no common tangents.

In our case:
\[ d = 2\sqrt{5} \approx 4.47 > 4 \quad (r_1 + r_2) \]
Thus, the circles have 4 common tangents.

### Final Answer
The number of common tangents of the circles is **4**.

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