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 determine the values of \(\lambda\) for which the two circles intersect at two distinct points, we need to analyze the properties of the circles given by their equations.

### Step 1: Identify the centers and radii of the circles.

1. The first circle is given by:
\[ (x - 2)^2 + (y + 3)^2 = \lambda^2 \]
- Center \(C_1 = (2, -3)\)
- Radius \(R_1 = \lambda\)

2. The second circle is given by:
\[ x^2 + y^2 - 4x + 4y - 1 = 0 \]
We need to rewrite this in standard form. Completing the square:
\[ (x^2 - 4x) + (y^2 + 4y) = 1 \]
Completing the square for \(x\):
\[ (x - 2)^2 - 4 \]
Completing the square for \(y\):
\[ (y + 2)^2 - 4 \]
Putting it all together:
\[ (x - 2)^2 + (y + 2)^2 - 8 = 1 \implies (x - 2)^2 + (y + 2)^2 = 9 \]
- Center \(C_2 = (2, -2)\)
- Radius \(R_2 = 3\)

### Step 2: Calculate the distance between the centers.

The distance \(d\) between the centers \(C_1\) and \(C_2\) is given by:
\[ d = \sqrt{(2 - 2)^2 + (-3 - (-2))^2} = \sqrt{0 + (-1)^2} = 1 \]

### Step 3: Apply the conditions for intersection.

For two circles to intersect at two distinct points, the following conditions must be satisfied:

1. The distance between the centers must be less than the sum of the radii:
\[ d < R_1 + R_2 \implies 1 < \lambda + 3 \]
Simplifying this gives:
\[ \lambda > -2 \]

2. The distance between the centers must be greater than the absolute difference of the radii:
\[ d > |R_1 - R_2| \implies 1 > |\lambda - 3| \]
This can be split into two inequalities:
\[ -1 < \lambda - 3 < 1 \]
Solving these inequalities:
- From \(-1 < \lambda - 3\):
\[ \lambda > 2 \]
- From \(\lambda - 3 < 1\):
\[ \lambda < 4 \]

### Step 4: Combine the inequalities.

Now we have two conditions:
1. \(\lambda > -2\)
2. \(2 < \lambda < 4\)

The intersection of these conditions gives:
\[ 2 < \lambda < 4 \]

### Conclusion

Thus, the values of \(\lambda\) for which the two circles intersect at two distinct points are:
\[ \boxed{(2, 4)} \]