If the equations `x^(2)+2lambdax+lambda^(2)+1=0`, `lambda in R` and `ax^(2)+bx+c=0` , where `a`, `b`, `c` are lengths of sides of triangle have a common root, then the possible range of values of `lambda` is

Options

A. `(0,2)`
B. `(sqrt(3),3)`
C. `(2sqrt(2),3sqrt(2))`
D. `(0,oo)`

Explanation

To solve the problem, we need to find the range of values of \(\lambda\) such that the equations \(x^2 + 2\lambda x + \lambda^2 + 1 = 0\) and \(ax^2 + bx + c = 0\) (where \(a\), \(b\), and \(c\) are the lengths of the sides of a triangle) have a common root.

### Step-by-step Solution:

1. **Identify the first equation**:
The first equation is:
\[ x^2 + 2\lambda x + \lambda^2 + 1 = 0 \]
This can be rewritten as:
\[ (x + \lambda)^2 + 1 = 0 \]
This indicates that the roots are complex and occur in conjugate pairs.

2. **Let the common root be \(r\)**:
Since both equations share a common root \(r\), we can substitute \(r\) into both equations.

3. **Set up the second equation**:
The second equation is:
\[ ax^2 + bx + c = 0 \]
Substituting \(r\) into this equation gives:
\[ ar^2 + br + c = 0 \]

4. **Express \(a\), \(b\), and \(c\) in terms of \(k\)**:
Since \(a\), \(b\), and \(c\) are the sides of a triangle, we can express them in terms of a common variable \(k\):
\[ a = k, \quad b = 2\lambda k, \quad c = (\lambda^2 + 1)k \]

5. **Substitute \(a\), \(b\), and \(c\) into the second equation**:
Replacing \(a\), \(b\), and \(c\) in the second equation gives:
\[ kr^2 + (2\lambda k)r + (\lambda^2 + 1)k = 0 \]
Dividing through by \(k\) (assuming \(k \neq 0\)):
\[ r^2 + 2\lambda r + (\lambda^2 + 1) = 0 \]

6. **Establish inequalities based on triangle properties**:
For \(a\), \(b\), and \(c\) to form a triangle, the triangle inequality must hold:
\[ a + b > c, \quad a + c > b, \quad b + c > a \]
Substituting the values:
- \(k + 2\lambda k > (\lambda^2 + 1)k\)
- \(k + (\lambda^2 + 1)k > 2\lambda k\)
- \(2\lambda k + (\lambda^2 + 1)k > k\)

7. **Simplify the inequalities**:
From the first inequality:
\[ 1 + 2\lambda > \lambda^2 + 1 \implies 2\lambda > \lambda^2 \implies \lambda^2 - 2\lambda < 0 \]
Factoring gives:
\[ \lambda(\lambda - 2) < 0 \]

8. **Determine the range of \(\lambda\)**:
The inequality \(\lambda(\lambda - 2) < 0\) indicates that \(\lambda\) must lie between the roots \(0\) and \(2\):
\[ 0 < \lambda < 2 \]

### Conclusion:
The possible range of values for \(\lambda\) is:
\[ \lambda \in (0, 2) \]