If the equation `x^(2 )+ 2x + 3 = 0` and `ax^(2) +bx+c=0, a, b, c in R`, have a common root, then a : b:c is

Options

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

Answer

To solve the problem, we need to find the ratio \( a : b : c \) given that the equations \( x^2 + 2x + 3 = 0 \) and \( ax^2 + bx + c = 0 \) have a common root.

### Step-by-Step Solution:

1. **Identify the first equation**:
The first equation is given as:
\[ x^2 + 2x + 3 = 0 \]

2. **Calculate the discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
For our first equation, \( a = 1, b = 2, c = 3 \):
\[ D = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8 \]
Since the discriminant is negative, this equation has imaginary roots.

3. **Find the roots of the first equation**:
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ x = \frac{-2 \pm \sqrt{-8}}{2 \cdot 1} = \frac{-2 \pm 2\sqrt{2}i}{2} = -1 \pm \sqrt{2}i \]
Thus, the roots are:
\[ x_1 = -1 + \sqrt{2}i \quad \text{and} \quad x_2 = -1 - \sqrt{2}i \]

4. **Assume a common root**:
Let’s assume that the common root of the second equation \( ax^2 + bx + c = 0 \) is \( x_1 = -1 + \sqrt{2}i \).

5. **Form the second equation**:
Since \( ax^2 + bx + c = 0 \) has the same root, we can express it in terms of its roots. The roots of the second equation can be expressed as:
\[ x = -1 + \sqrt{2}i \quad \text{and} \quad x = -1 - \sqrt{2}i \]
Therefore, we can write the second equation as:
\[ a(x - (-1 + \sqrt{2}i))(x - (-1 - \sqrt{2}i)) = 0 \]
Expanding this:
\[ a[(x + 1 - \sqrt{2}i)(x + 1 + \sqrt{2}i)] = 0 \]
This can be simplified using the difference of squares:
\[ a[(x + 1)^2 - (\sqrt{2}i)^2] = 0 \]
\[ = a[(x + 1)^2 + 2] = 0 \]
Expanding \( (x + 1)^2 + 2 \):
\[ = a[x^2 + 2x + 1 + 2] = a[x^2 + 2x + 3] \]
Thus, we have:
\[ ax^2 + 2ax + 3a = 0 \]

6. **Compare coefficients**:
By comparing coefficients of \( ax^2 + bx + c \) with \( x^2 + 2x + 3 \), we get:
\[ b = 2a \quad \text{and} \quad c = 3a \]

7. **Find the ratio \( a : b : c \)**:
We can express \( a : b : c \) as:
\[ a : 2a : 3a = 1 : 2 : 3 \]

### Final Answer:
Thus, the ratio \( a : b : c \) is:
\[ \boxed{1 : 2 : 3} \]