Let `a ,b , c` be the sides of a triangle, where `a!=b!=c` and `lambda in R` . If the roots of the equation `x^2+2(a+b+c)x+3lambda(a b+b c+c a)=0` are real. Then a.`lambda<4/3` b. `lambda>5/3` c. `lambda in (1/3,5/3)` d. `lambda in (4/3,5/3)`

Options

A. `lamda lt 4/3`
B. `lamda lt 5/3.`
C. `l epsilon(1/3,5/3)`
D. `lamda epsilon (4/3,5/3)`

Explanation

To solve the problem, we need to analyze the quadratic equation given and determine the conditions under which its roots are real. The equation is:

\[ x^2 + 2(a + b + c)x + 3\lambda(ab + bc + ca) = 0 \]

### Step 1: Identify the discriminant
For the roots of a quadratic equation \( ax^2 + bx + c = 0 \) to be real, the discriminant must be non-negative. The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]
...