If the equations `ax^2 + bx + c = 0 and x^2 + x + 1= 0` has one common root then `a : b : c` is equal to

Options

A. `1:1:1`
B. `1:2:3`
C. `2:3:1`
D. ` 3: 2: 1`

Answer

To solve the problem, we need to find the ratio \( a : b : c \) given that the equations \( ax^2 + bx + c = 0 \) and \( x^2 + x + 1 = 0 \) have one common root.

### Step 1: Identify the roots of the second equation
The second equation is:
\[ x^2 + x + 1 = 0 \]
We can find the roots using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 1, c = 1 \). The discriminant is:
\[ b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]
Since the discriminant is negative, the roots are complex and can be expressed as:
\[ x = \frac{-1 \pm i\sqrt{3}}{2} \]
Let the roots be:
\[ \alpha = \frac{-1 + i\sqrt{3}}{2}, \quad \beta = \frac{-1 - i\sqrt{3}}{2} \]

### Step 2: Set up the first equation with a common root
Let’s assume that one of the roots of the first equation \( ax^2 + bx + c = 0 \) is \( \alpha \). The sum and product of the roots of the first equation can be expressed as follows:
- Sum of roots: \( -\frac{b}{a} \)
- Product of roots: \( \frac{c}{a} \)

### Step 3: Relate the roots of both equations
Since the first equation has one common root \( \alpha \) and the other root can be denoted as \( r \), we have:
- For the first equation:
- Sum of roots: \( \alpha + r = -\frac{b}{a} \)
- Product of roots: \( \alpha r = \frac{c}{a} \)

- For the second equation:
- The sum of roots is \( \alpha + \beta = -1 \)
- The product of roots is \( \alpha \beta = 1 \)

### Step 4: Equate the sums and products
From the sum of roots:
\[ \alpha + r = -\frac{b}{a} \]
Since \( \alpha + \beta = -1 \), we can write:
\[ r = -\frac{b}{a} - \alpha \]

From the product of roots:
\[ \alpha r = \frac{c}{a} \]
Substituting \( r \) gives:
\[ \alpha \left(-\frac{b}{a} - \alpha\right) = \frac{c}{a} \]

### Step 5: Solve for \( a, b, c \)
From the above equations, we can derive relationships between \( a, b, c \):
1. From the sum of roots, we have:
\[ -\frac{b}{a} = -1 \implies b = a \]
2. From the product of roots, we have:
\[ \alpha r = 1 \implies c = a \]

### Step 6: Find the ratio
Thus, we have:
\[ a = b = c \]
This implies the ratio:
\[ a : b : c = 1 : 1 : 1 \]

### Final Answer
The ratio \( a : b : c \) is \( 1 : 1 : 1 \).