Let `a,b,c` be the sides of a triangle. No two of them are equal and `lambda in R` If the roots of the equation `x^2+2(a+b+c)x+3lambda(ab+bc+ca)=0` are real, then (a) `lambda < 4/3` (b) `lambda > 5/3` (c) `lambda in (1/5,5/3)` (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 given problem, we need to analyze the quadratic equation provided 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 coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we have:
- \( a = 1 \)
- \( b = 2(a+b+c) \)
- \( c = 3\lambda(ab + bc + ca) \)

### Step 2: Apply the condition for real roots
For the roots of the quadratic equation to be real, the discriminant \( D \) must be greater than or equal to zero. The discriminant \( D \) is given by:

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

Substituting the coefficients:

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

### Step 3: Simplify the discriminant
Calculating \( D \):

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

### Step 4: Set the discriminant greater than or equal to zero
For the roots to be real:

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

### Step 5: Rearranging the inequality
Rearranging gives:

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

Dividing both sides by 4:

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

### Step 6: Use the triangle inequality
Since \( a, b, c \) are the sides of a triangle, we can apply the triangle inequality. We know:

\[ a^2 + b^2 + c^2 \geq ab + bc + ca \]

Thus, we can express \( (a+b+c)^2 \) in terms of \( ab + bc + ca \):

\[ (a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

### Step 7: Substitute the inequality
From the triangle inequality, we have:

\[ a^2 + b^2 + c^2 < 2(ab + bc + ca) \]

This leads to:

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

### Step 8: Substitute back into the inequality
Now substituting back into our inequality:

\[ 3(ab + bc + ca) \geq 3\lambda(ab + bc + ca) \]

Dividing both sides by \( ab + bc + ca \) (which is positive since \( a, b, c \) are sides of a triangle):

\[ 3 \geq 3\lambda \]

### Step 9: Solve for \( \lambda \)
Dividing by 3 gives:

\[ 1 \geq \lambda \]

Thus:

\[ \lambda \leq 1 \]

### Step 10: Analyze the options
Given the options:
(a) \( \lambda < \frac{4}{3} \)
(b) \( \lambda > \frac{5}{3} \)
(c) \( \lambda \in (1/5, 5/3) \)
(d) \( \lambda \in (4/3, 5/3) \)

The only option that satisfies \( \lambda \leq 1 \) is:

**Correct option: (a) \( \lambda < \frac{4}{3} \)**