Statement-1 : Let a quadratic equation has a root 3 - 9i then the sum of roots is 6.
and
Statement -2 : If one root of ` ax^(2) +bx +c=0 ,ane 0, a,b,c in R " is "alpha+ ibeta, alpha,beta in R` then the other roots must be `alpha-ibeta`

Options

A. Statement -1 is True, Statement -2 is True, Statement -2 is a correct explanation for statement -19
B. Statement -1 is True, Statement -2 is True , Statement -2 is NOT a correct explanation for Statement -19
C. Statement -1 is True, Statement -2 is False
D. Statement -1 is Flase, Statement -2 is True

Answer

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyze Statement 1
Let’s consider the first statement: "Let a quadratic equation has a root \(3 - 9i\). Then the sum of roots is 6."

1. **Identify the given root**: The root given is \(3 - 9i\).
2. **Conjugate root theorem**: For quadratic equations with real coefficients, if one root is complex (like \(3 - 9i\)), the other root must be its conjugate, which is \(3 + 9i\).
3. **Calculate the sum of the roots**:
\[ \text{Sum of roots} = (3 - 9i) + (3 + 9i) = 3 + 3 + (-9i + 9i) = 6 + 0i = 6 \]
Thus, the sum of the roots is indeed 6.

### Step 2: Analyze Statement 2
Now let’s consider the second statement: "If one root of \(ax^2 + bx + c = 0\) (where \(a \neq 0\) and \(a, b, c \in \mathbb{R}\)) is \(\alpha + i\beta\) (where \(\alpha, \beta \in \mathbb{R}\)), then the other root must be \(\alpha - i\beta\)."

1. **Identify the given root**: The root is \(\alpha + i\beta\).
2. **Conjugate root theorem**: As stated earlier, since the coefficients of the quadratic equation are real, the other root must be the conjugate of the given root, which is \(\alpha - i\beta\).
3. **Conclusion**: This statement is correct because it follows the conjugate root theorem.

### Final Conclusion
Both statements are correct:
- Statement 1 is correct because we calculated the sum of the roots and found it to be 6.
- Statement 2 is correct because it correctly describes the relationship between the roots of a quadratic equation with real coefficients.

Thus, the answer is that both statements are true, and Statement 2 provides a correct explanation for Statement 1.