Consider three circles `C_(1), C_(2)` and `C_(3)` as given below:
`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-1: If the circle `C_(3)` intersects `C_(1)` orthogonally , then `C_(2)` does not represent a circle.
Statement-2: If the circle `C_(3)` intersects `C_(2)` orthogonally, then `C_(2)` and `C_(3)` have equal radii.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Explanation

To solve the problem, we need to analyze the two statements provided regarding the circles \( C_1 \), \( C_2 \), and \( C_3 \).

### Step 1: Analyze Statement 1
**Statement 1**: If the circle \( C_3 \) intersects \( C_1 \) orthogonally, then \( C_2 \) does not represent a circle.

1. **Equations of the Circles**:
- \( C_1: x^2 + y^2 + 2x - 2y + p = 0 \)
- \( C_2: x^2 + y^2 - 2x + 2y - p = 0 \)
- \( C_3: x^2 + y^2 = p^2 \)

2. **Orthogonality Condition**:
Two circles \( C_1 \) and \( C_2 \) intersect orthogonally if:
\[ g_1 g_2 + f_1 f_2 = c_1 + c_2 \]
where \( g, f, c \) are the coefficients from the general circle equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \).

3. **Identifying Coefficients**:
- For \( C_1 \):
- \( g_1 = 1 \), \( f_1 = -1 \), \( c_1 = p \)
- For \( C_3 \):
- \( g_3 = 0 \), \( f_3 = 0 \), \( c_3 = -p^2 \)
- For \( C_2 \):
- \( g_2 = -1 \), \( f_2 = 1 \), \( c_2 = -p \)

4. **Substituting into the Orthogonality Condition**:
\[ (1)(-1) + (-1)(1) = p + (-p^2) \]
Simplifying gives:
\[ -1 - 1 = p - p^2 \]
\[ -2 = p - p^2 \]
Rearranging:
\[ p^2 - p - 2 = 0 \]

5. **Finding Roots**:
Using the quadratic formula:
\[ p = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} \]
\[ p = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm 3}{2} \]
Thus, \( p = 2 \) or \( p = -1 \).

6. **Checking Validity of \( p \)**:
- If \( p = -1 \), \( C_2 \) becomes:
\[ x^2 + y^2 - 2x + 2y + 1 = 0 \]
This does represent a circle.
- If \( p = 2 \), \( C_2 \) becomes:
\[ x^2 + y^2 - 2x + 2y - 2 = 0 \]
This also represents a circle.

Thus, Statement 1 is **false** because \( C_2 \) can represent a circle for both valid values of \( p \).

### Step 2: Analyze Statement 2
**Statement 2**: If the circle \( C_3 \) intersects \( C_2 \) orthogonally, then \( C_2 \) and \( C_3 \) have equal radii.

1. **Using the Orthogonality Condition Again**:
For \( C_2 \) and \( C_3 \):
\[ g_2 g_3 + f_2 f_3 = c_2 + c_3 \]
Substituting the coefficients:
\[ (-1)(0) + (1)(0) = -p + (-p^2) \]
Simplifying gives:
\[ 0 = -p - p^2 \]
Rearranging:
\[ p^2 + p = 0 \]
Factoring:
\[ p(p + 1) = 0 \]
Thus, \( p = 0 \) or \( p = -1 \).

2. **Checking Validity of \( p \)**:
- If \( p = 0 \), \( C_3 \) does not represent a circle.
- If \( p = -1 \), both \( C_2 \) and \( C_3 \) have the same radius.

Thus, Statement 2 is **true**.

### Conclusion
- Statement 1 is **false**.
- Statement 2 is **true**.