Consider circles
`C_(1): x^(2) +y^(2) +2x - 2y +p = 0`
`C_(2): x^(2) +y^(2) - 2x +2y - p = 0`
`C_(3): x^(2) +y^(2) = p^(2)`
Statement-I: If the circle `C_(3)` intersects `C_(1)` orthogonally then `C_(2)` does not represent a circle
Statement-II: If the circle `C_(3)` intersects `C_(2)` orthogonally then `C_(2)` and `C_(3)` have equal radii Then which of the following is true?

Options

A. statement II is false and statement I is true
B. statement I is false and statement II is true
C. both the statements are false
D. both the statements are true

Explanation

To solve the problem, we need to analyze the given circles and the statements regarding their intersections.

### Step 1: Identify the equations of the circles
The equations of the circles are given as:
- Circle \( C_1: x^2 + y^2 + 2x - 2y + p = 0 \)
- Circle \( C_2: x^2 + y^2 - 2x + 2y - p = 0 \)
- Circle \( C_3: x^2 + y^2 = p^2 \)

### Step 2: Rewrite the equations in standard form
To analyze the circles, we can rewrite the equations in standard form \( (x - h)^2 + (y - k)^2 = r^2 \).

1. **For Circle \( C_1 \)**:
\[ x^2 + y^2 + 2x - 2y + p = 0 \implies (x + 1)^2 + (y - 1)^2 = 1 - p \]
- Center: \( (-1, 1) \)
- Radius: \( \sqrt{1 - p} \)

2. **For Circle \( C_2 \)**:
\[ x^2 + y^2 - 2x + 2y - p = 0 \implies (x - 1)^2 + (y + 1)^2 = 1 + p \]
- Center: \( (1, -1) \)
- Radius: \( \sqrt{1 + p} \)

3. **For Circle \( C_3 \)**:
\[ x^2 + y^2 = p^2 \implies (x - 0)^2 + (y - 0)^2 = p^2 \]
- Center: \( (0, 0) \)
- Radius: \( p \)

### Step 3: Determine conditions for orthogonal intersection
Two circles intersect orthogonally if the sum of the products of their radii equals the square of the distance between their centers.

1. **For \( C_1 \) and \( C_3 \)**:
- Distance between centers:
\[ d = \sqrt{((-1) - 0)^2 + (1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \]
- Condition for orthogonality:
\[ (1 - p) + p^2 = 2 \]
Rearranging gives:
\[ p^2 - p - 1 = 0 \]

2. **For \( C_2 \) and \( C_3 \)**:
- Distance between centers:
\[ d = \sqrt{(1 - 0)^2 + ((-1) - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \]
- Condition for orthogonality:
\[ (1 + p) + p^2 = 2 \]
Rearranging gives:
\[ p^2 + p - 1 = 0 \]

### Step 4: Analyze the statements
- **Statement I**: If \( C_3 \) intersects \( C_1 \) orthogonally, then \( C_2 \) does not represent a circle.
- From our analysis, we found that \( C_2 \) has a valid radius \( \sqrt{1 + p} \). Therefore, this statement is **false**.

- **Statement II**: If \( C_3 \) intersects \( C_2 \) orthogonally, then \( C_2 \) and \( C_3 \) have equal radii.
- From our analysis, we found that both circles can have equal radii under certain conditions. Therefore, this statement is **true**.

### Conclusion
Based on the analysis:
- Statement I is false.
- Statement II is true.

Thus, the correct option is:
**Option 2: Statement 1 is false and Statement 2 is true.**