If the two circles `(x-2)^(2)+(y+3)^(2) =lambda^(2)` and `x^(2)+y^(2) -4x +4y-1=0` intersect in two distinct points then :

Options

A. `lambda lt -2`
B. `-2 lt lambda lt 4`
C. `2 lt lambda lt 4`
D. `lambda = -2 `

Explanation

To solve the problem of determining the values of \(\lambda\) for which the two circles intersect at two distinct points, we will follow these steps:

### Step 1: Identify the equations of the circles
The first circle is given by:
\[ (x-2)^2 + (y+3)^2 = \lambda^2 \]
This is already in standard form, where the center \(C_1\) is \((2, -3)\) and the radius \(r_1 = \lambda\).

The second circle is given by:
\[ x^2 + y^2 - 4x + 4y - 1 = 0 \]
We will rewrite this in standard form.

### Step 2: Rewrite the second circle in standard form
To rewrite the second circle, we complete the square for both \(x\) and \(y\):
\[ x^2 - 4x + y^2 + 4y = 1 \]
Completing the square for \(x\):
\[ (x-2)^2 - 4 \]
Completing the square for \(y\):
\[ (y+2)^2 - 4 \]
Substituting back, we have:
\[ (x-2)^2 - 4 + (y+2)^2 - 4 = 1 \]
This simplifies to:
\[ (x-2)^2 + (y+2)^2 = 9 \]
Thus, the center \(C_2\) is \((2, -2)\) and the radius \(r_2 = 3\).

### Step 3: Find the distance between the centers of the circles
The distance \(d\) between the centers \(C_1(2, -3)\) and \(C_2(2, -2)\) is calculated as follows:
\[ d = \sqrt{(2-2)^2 + (-3 - (-2))^2} = \sqrt{0 + (-1)^2} = 1 \]

### Step 4: Apply the intersection condition for circles
For two circles to intersect at two distinct points, the following conditions must hold:
1. \(d < r_1 + r_2\)
2. \(d > |r_1 - r_2|\)

Substituting the known values:
1. \(1 < \lambda + 3\)
2. \(1 > |\lambda - 3|\)

### Step 5: Solve the inequalities
**From the first inequality:**
\[ 1 < \lambda + 3 \implies \lambda > -2 \]

**From the second inequality:**
\[ 1 > |\lambda - 3| \implies -1 < \lambda - 3 < 1 \]
This gives us two inequalities:
1. \(\lambda - 3 > -1 \implies \lambda > 2\)
2. \(\lambda - 3 < 1 \implies \lambda < 4\)

### Step 6: Combine the inequalities
Combining the results, we have:
\[ 2 < \lambda < 4 \]

### Conclusion
Thus, the values of \(\lambda\) for which the two circles intersect at two distinct points are:
\[ \lambda \in (2, 4) \]