Let C be the circle `x^2+y^2-4x-4y-1=0.` The number of 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 given circle 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 equation of the circle is given as:
\[ x^2 + y^2 - 4x - 4y - 1 = 0 \]
We can rearrange this equation to find the center and radius of the circle. Completing the square for both \(x\) and \(y\):

1. For \(x\):
\[ x^2 - 4x = (x-2)^2 - 4 \]
2. For \(y\):
\[ y^2 - 4y = (y-2)^2 - 4 \]

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

### Step 2: Identify the rectangle's vertices
The rectangle is defined by the lines:
- \(x = 2\)
- \(x = 5\)
- \(y = -1\)
- \(y = 5\)

The vertices of the rectangle can be found by combining these lines:
1. \((2, -1)\)
2. \((2, 5)\)
3. \((5, -1)\)
4. \((5, 5)\)

### Step 3: Find intersection points with the sides of the rectangle
Now we will check the intersections of the circle with each side of the rectangle.

#### Side 1: \(x = 2\)
Substituting \(x = 2\) into the circle's equation:
\[ (2-2)^2 + (y-2)^2 = 9 \implies (y-2)^2 = 9 \]
This gives:
\[ y - 2 = 3 \quad \text{or} \quad y - 2 = -3 \]
Thus:
\[ y = 5 \quad \text{or} \quad y = -1 \]
So, the points of intersection on this side are \((2, 5)\) and \((2, -1)\).

#### Side 2: \(x = 5\)
Substituting \(x = 5\) into the circle's equation:
\[ (5-2)^2 + (y-2)^2 = 9 \implies 9 + (y-2)^2 = 9 \]
This simplifies to:
\[ (y-2)^2 = 0 \implies y = 2 \]
So, the point of intersection on this side is \((5, 2)\).

#### Side 3: \(y = -1\)
Substituting \(y = -1\) into the circle's equation:
\[ (x-2)^2 + (-1-2)^2 = 9 \implies (x-2)^2 + 9 = 9 \]
This simplifies to:
\[ (x-2)^2 = 0 \implies x = 2 \]
So, the point of intersection on this side is \((2, -1)\) (already counted).

#### Side 4: \(y = 5\)
Substituting \(y = 5\) into the circle's equation:
\[ (x-2)^2 + (5-2)^2 = 9 \implies (x-2)^2 + 9 = 9 \]
This simplifies to:
\[ (x-2)^2 = 0 \implies x = 2 \]
So, the point of intersection on this side is \((2, 5)\) (already counted).

### Step 4: Count the unique intersection points
The unique points of intersection are:
1. \((2, 5)\)
2. \((2, -1)\)
3. \((5, 2)\)

Thus, the total number of unique intersection points is \(3\).

### Final Answer:
The number of points common to the circle and the sides of the rectangle is **3**.