The number of common tangents of the circles `x^2+y^2−2x−1=0` and `x^2+y^2−2y−7=0`

Answer

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

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

1. For the first circle:
\[ x^2 + y^2 - 2x - 1 = 0 \implies (x^2 - 2x + 1) + y^2 = 1 \implies (x - 1)^2 + y^2 = 1 \]
This gives us:
- Center \(C_1(1, 0)\)
- Radius \(r_1 = 1\)

2. For the second circle:
\[ x^2 + y^2 - 2y - 7 = 0 \implies x^2 + (y^2 - 2y + 1) = 8 \implies x^2 + (y - 1)^2 = 8 \]
This gives us:
- Center \(C_2(0, 1)\)
- Radius \(r_2 = \sqrt{8} = 2\sqrt{2}\)

### Step 2: Calculate the distance between the centers of the circles
Next, we calculate the distance \(d\) between the centers \(C_1\) and \(C_2\):
\[ d = \sqrt{(1 - 0)^2 + (0 - 1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Compare the distance with the sum and difference of the radii
Now, we need to compare the distance \(d\) with the sum and difference of the radii \(r_1\) and \(r_2\):
- Sum of the radii: \(r_1 + r_2 = 1 + 2\sqrt{2}\)
- Difference of the radii: \(r_2 - r_1 = 2\sqrt{2} - 1\)

### Step 4: Determine the relationship between \(d\), \(r_1\), and \(r_2\)
We need to check the following conditions:
1. If \(d > r_1 + r_2\): 2 external tangents
2. If \(d = r_1 + r_2\): 1 external tangent
3. If \(r_2 - r_1 < d < r_1 + r_2\): 2 common tangents
4. If \(d = r_2 - r_1\): 1 internal tangent
5. If \(d < r_2 - r_1\): No common tangents

Calculating:
- \(d = \sqrt{2}\)
- \(r_1 + r_2 = 1 + 2\sqrt{2}\) (which is greater than \(\sqrt{2}\))
- \(r_2 - r_1 = 2\sqrt{2} - 1\) (which is also greater than \(\sqrt{2}\))

### Conclusion
Since \(d\) (which is \(\sqrt{2}\)) is less than both \(r_1 + r_2\) and \(r_2 - r_1\), we conclude that the circles intersect internally, and thus they have only one common tangent.

### Final Answer
The number of common tangents of the circles is **1**.