Let `a_(1)x^(2)+b_(1)x + c_(1)=0` and `a_(2)x^(2)+b_(2)x+c_(2)=0` be the quaratic such that `Delta_(1)=b_(1)^(2)-4a_(1)c_(1)` and `Delta_(2)=b_(2)^(2)-4a_(2)c_(2)`.
Statement 1 : Atleast one of the given equations have imaginary roots if and only if `Delta_(1)+Delta_(2)lt 0`.
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Statement 2 : If `Delta_(1)+Delta_(2)lt 0`, then atleast one of the `Delta_(1)` and `Delta_(2)` is negative.

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 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

Answer

To solve the problem, we need to analyze the two statements given regarding the quadratic equations and their discriminants.

### Step 1: Understand the Quadratic Equations
We have two quadratic equations:
1. \( a_1 x^2 + b_1 x + c_1 = 0 \) with discriminant \( \Delta_1 = b_1^2 - 4a_1c_1 \)
2. \( a_2 x^2 + b_2 x + c_2 = 0 \) with discriminant \( \Delta_2 = b_2^2 - 4a_2c_2 \)

### Step 2: Analyze Statement 1
**Statement 1**: At least one of the given equations has imaginary roots if and only if \( \Delta_1 + \Delta_2 < 0 \).

- **Imaginary Roots**: A quadratic equation has imaginary roots if its discriminant is less than zero. Therefore:
- For the first equation: \( \Delta_1 < 0 \) implies imaginary roots.
- For the second equation: \( \Delta_2 < 0 \) implies imaginary roots.

- If \( \Delta_1 + \Delta_2 < 0 \), it means that the sum of the discriminants is negative. This can occur if:
- Both \( \Delta_1 \) and \( \Delta_2 \) are negative, or
- One is negative, and the other is less positive (but still positive).

Thus, if \( \Delta_1 + \Delta_2 < 0 \), at least one of the discriminants must be negative, indicating that at least one of the equations has imaginary roots.

### Step 3: Analyze Statement 2
**Statement 2**: If \( \Delta_1 + \Delta_2 < 0 \), then at least one of \( \Delta_1 \) and \( \Delta_2 \) is negative.

- This statement is straightforward. If the sum of two numbers (in this case, the discriminants) is negative, at least one of those numbers must be negative. Therefore, if \( \Delta_1 + \Delta_2 < 0 \), it follows that at least one of \( \Delta_1 \) or \( \Delta_2 \) must be negative.

### Conclusion:
- **Statement 1** is true because at least one of the equations must have imaginary roots if \( \Delta_1 + \Delta_2 < 0 \).
- **Statement 2** is also true because if the sum of the discriminants is negative, at least one must be negative.

Thus, the correct answer is that both statements are true.