Let a,b and c be the sides of a triangle. No two of them are equal and `lambda` `epsilon`R. If the roots of the equation
`x^2+2(a+b+c)x+3lambda(ab+bc+ca)=0`
are real, then

Options

A. `lambdalt4/3`
B. `lambdagt5/3`
C. `1/3ltlambdalt5/3`
D. `4/3ltlambdalt5/3`

Answer

To solve the problem, we need to analyze the given quadratic equation and the conditions for its roots to be real. The equation is:

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

### Step 1: Identify the discriminant condition
For the roots of a quadratic equation \( ax^2 + bx + c = 0 \) to be real, the discriminant must be greater than or equal to zero. The discriminant \( D \) is given by:

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

In our case, \( a = 1 \), \( b = 2(a+b+c) \), and \( c = 3\lambda(ab + bc + ca) \).

### Step 2: Calculate the discriminant
Substituting the values into the discriminant formula, we have:

\[ D = [2(a+b+c)]^2 - 4 \cdot 1 \cdot 3\lambda(ab + bc + ca) \]

This simplifies to:

\[ D = 4(a+b+c)^2 - 12\lambda(ab + bc + ca) \]

### Step 3: Set the discriminant greater than zero
For the roots to be real, we need:

\[ 4(a+b+c)^2 - 12\lambda(ab + bc + ca) > 0 \]

### Step 4: Rearranging the inequality
Rearranging gives us:

\[ 4(a+b+c)^2 > 12\lambda(ab + bc + ca) \]

Dividing both sides by 4, we get:

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

### Step 5: Express the inequality in terms of lambda
Now, we can express this in terms of \( \lambda \):

\[ \lambda < \frac{(a+b+c)^2}{3(ab + bc + ca)} \]

### Step 6: Analyze the triangle inequality
Since \( a, b, c \) are the sides of a triangle, we can use the triangle inequalities. The triangle inequalities state that:

1. \( a + b > c \)
2. \( b + c > a \)
3. \( c + a > b \)

From these inequalities, we can derive that:

\[ a + b + c > 2\max(a, b, c) \]

### Step 7: Substitute and simplify
Using the properties of the triangle, we can analyze the expression \( ab + bc + ca \) and find bounds for \( \lambda \).

### Conclusion
After analyzing the inequalities and substituting the triangle properties, we find that:

\[ \lambda < \frac{(a+b+c)^2}{3(ab + bc + ca)} \]

This leads us to conclude that \( \lambda \) must be less than \( \frac{4}{3} \) based on the derived inequalities.

### Final Answer
Thus, the condition for \( \lambda \) is:

\[ \lambda < \frac{4}{3} \]