Circle `(x + 1)^2 + (y + 2)^2 = r^2` & `x^2 + y^2 - 4x - 4y + 4 = 0`
Cuts each other at two different points then value of r is

Options

A. `1/2 lt r lt 7`
B. `0 lt r lt 7`
C. `3 lt r lt 7`
D. `5 lt r lt 9`

Answer

To solve the problem, we need to find the value of \( r \) such that the circle given by the equation \( (x + 1)^2 + (y + 2)^2 = r^2 \) intersects with the circle represented by the equation \( x^2 + y^2 - 4x - 4y + 4 = 0 \) at two distinct points.

### Step 1: Rewrite the second circle's equation in standard form
The given equation of the second circle is:
\[ x^2 + y^2 - 4x - 4y + 4 = 0 \]
We can rearrange this to find the center and radius. First, we complete the square for \( x \) and \( y \):

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

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

Substituting these back into the equation gives:
\[ (x - 2)^2 - 4 + (y - 2)^2 - 4 + 4 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 2)^2 - 4 = 0 \]
Thus, we have:
\[ (x - 2)^2 + (y - 2)^2 = 4 \]
This indicates that the center of the second circle is \( (2, 2) \) and its radius \( R = 2 \).

### Step 2: Determine the distance between the centers of the circles
The center of the first circle \( (x + 1)^2 + (y + 2)^2 = r^2 \) is \( (-1, -2) \). The center of the second circle is \( (2, 2) \).

Now, we calculate the distance \( d \) between the two centers:
\[ d = \sqrt{(2 - (-1))^2 + (2 - (-2))^2} = \sqrt{(2 + 1)^2 + (2 + 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Apply the condition for intersection
For two circles to intersect at two distinct points, the following condition must hold:
\[ |R - r| < d < R + r \]
Where:
- \( R = 2 \) (radius of the second circle)
- \( r \) (radius of the first circle)
- \( d = 5 \)

Substituting into the inequalities:
1. \( |2 - r| < 5 \)
2. \( 5 < 2 + r \)

### Step 4: Solve the inequalities
From \( |2 - r| < 5 \):
- This splits into two cases:
1. \( 2 - r < 5 \) which gives \( r > -3 \) (not useful since radius can't be negative)
2. \( 2 - r > -5 \) which gives \( r < 7 \)

From \( 5 < 2 + r \):
- Rearranging gives \( r > 3 \).

### Step 5: Combine the results
From the inequalities:
- \( 3 < r < 7 \)

Thus, the value of \( r \) must be greater than 3 and less than 7 for the circles to intersect at two distinct points.

### Conclusion
The value of \( r \) is in the range \( (3, 7) \).