Let C be the circle `x^2+y^2-4x-4y-1=0.` The number oof points common to C and the sides of the rectangle determined by the lines `x=2,x=5,y=-1 and y=5` equal to

Answer

To solve the problem, we need to find the number of points where the circle \( C \) intersects with the sides of the rectangle defined by the lines \( x = 2, x = 5, y = -1, \) and \( y = 5 \).

### Step 1: Rewrite the equation of the circle
The given equation of the circle is:
\[ x^2 + y^2 - 4x - 4y - 1 = 0 \]
We can rearrange this into standard form by completing the square.

### Step 2: Complete the square
1. For \( x \):
\[ x^2 - 4x \rightarrow (x - 2)^2 - 4 \]
2. For \( y \):
\[ y^2 - 4y \rightarrow (y - 2)^2 - 4 \]

Substituting back into the equation, we have:
\[ (x - 2)^2 - 4 + (y - 2)^2 - 4 - 1 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 2)^2 = 9 \]
This shows that the circle is centered at \( (2, 2) \) with a radius of \( 3 \).

### Step 3: Find intersections with the lines \( x = 2 \) and \( x = 5 \)
1. **For \( x = 2 \)**:
Substitute \( x = 2 \) into the circle's equation:
\[ (2 - 2)^2 + (y - 2)^2 = 9 \implies (y - 2)^2 = 9 \]
This gives:
\[ y - 2 = 3 \quad \text{or} \quad y - 2 = -3 \implies y = 5 \quad \text{or} \quad y = -1 \]
So, the points of intersection are \( (2, 5) \) and \( (2, -1) \).

2. **For \( x = 5 \)**:
Substitute \( x = 5 \) into the circle's equation:
\[ (5 - 2)^2 + (y - 2)^2 = 9 \implies 9 + (y - 2)^2 = 9 \]
This simplifies to:
\[ (y - 2)^2 = 0 \implies y - 2 = 0 \implies y = 2 \]
So, the point of intersection is \( (5, 2) \).

### Step 4: Find intersections with the lines \( y = -1 \) and \( y = 5 \)
1. **For \( y = -1 \)**:
Substitute \( y = -1 \) into the circle's equation:
\[ (x - 2)^2 + (-1 - 2)^2 = 9 \implies (x - 2)^2 + 9 = 9 \]
This simplifies to:
\[ (x - 2)^2 = 0 \implies x - 2 = 0 \implies x = 2 \]
So, the point of intersection is \( (2, -1) \) (already counted).

2. **For \( y = 5 \)**:
Substitute \( y = 5 \) into the circle's equation:
\[ (x - 2)^2 + (5 - 2)^2 = 9 \implies (x - 2)^2 + 9 = 9 \]
This simplifies to:
\[ (x - 2)^2 = 0 \implies x - 2 = 0 \implies x = 2 \]
So, the point of intersection is \( (2, 5) \) (already counted).

### Step 5: Count the unique intersection points
From the calculations, we have:
- From \( x = 2 \): \( (2, 5) \) and \( (2, -1) \)
- From \( x = 5 \): \( (5, 2) \)
- From \( y = -1 \): \( (2, -1) \) (already counted)
- From \( y = 5 \): \( (2, 5) \) (already counted)

Thus, the unique points of intersection are:
1. \( (2, 5) \)
2. \( (2, -1) \)
3. \( (5, 2) \)

### Conclusion
The total number of unique points common to the circle and the sides of the rectangle is \( 3 \).