Let a, b, c be real numbers such that `ax^(2) + bx + c = 0 and x^(2) + x + 1 = 0` have a common root.
Statement-1: a = b = c
Staement-2: Two quadratic equations with real coefficients cannot have only one imainary root common.

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.

Explanation

To solve the problem, we need to analyze the two quadratic equations given:

1. \( ax^2 + bx + c = 0 \)
2. \( x^2 + x + 1 = 0 \)

We are told that these two equations have a common root. Let's follow the steps to find out what this implies about the coefficients \( a, b, \) and \( c \).

### Step 1: Find the roots of the second equation
The roots of the equation \( x^2 + x + 1 = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 1 \), and \( c = 1 \). Plugging these values into the formula:
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} \]
This simplifies to:
\[ x = \frac{-1 \pm i\sqrt{3}}{2} \]
Thus, the roots of the equation \( x^2 + x + 1 = 0 \) are \( \frac{-1 + i\sqrt{3}}{2} \) and \( \frac{-1 - i\sqrt{3}}{2} \).

### Step 2: Understand the implications of a common root
Since \( ax^2 + bx + c = 0 \) has a common root with \( x^2 + x + 1 = 0 \), it must also have imaginary roots. The common root must be one of the roots we found above, either \( \frac{-1 + i\sqrt{3}}{2} \) or \( \frac{-1 - i\sqrt{3}}{2} \).

### Step 3: Equate the two equations
For the two equations to have the same roots, the coefficients \( a, b, c \) must be proportional to the coefficients of \( x^2 + x + 1 \). Therefore, we can write:
\[ \frac{a}{1} = \frac{b}{1} = \frac{c}{1} \]
This implies that \( a = b = c \).

### Step 4: Validate the statements
- **Statement 1**: \( a = b = c \) is true based on our analysis.
- **Statement 2**: It states that two quadratic equations with real coefficients cannot have only one imaginary root in common. This is also true because if they have one common imaginary root, they must have both roots in common (since complex roots come in conjugate pairs).

### Conclusion
Both statements are true.