Question

If a,b,c are the sides of a triangle ABC such that x^2 - 2(a + b + c)x + 3λ(ab + bc + ca) = 0 has real roots,

Answer 1

please write clearly all variable

Answer 2

Answer:

we know that difference of two sides of a triangle

is less then the third side.

| a - b | < c → a² + b² - 2ab < c² ........ (1)

| b - c | < a → b² + c² - 2bc < a² ........ (2)

| c - a | < b → c² + a² - 2ca < b² ........ (3)

Now, Adding eq (1) and (2) and (3)

→ 2a² + 2b² + 2c² - 2(ab + bc + ca) < a² + b² + c²

→ a² + b² + c² < 2ab + 2bc + 2ca ........ (A)

Given that the equation has real roots. So, the discriminant (∆) ≥ 0

→ b² - 4ac ≥ 0

4(a + b + c)² - 12λ(ab + bc + ca) ≥ 0

(a + b + c)² ≥ 3λ(ab + bc + ca) ≥ 0

a² + b² + c² ≥ (3λ - 2) (ab + bc + ca) ........ (B)

From eq (A) and (B)

2(ab + bc + ca) > (3λ - 2) (ab + bc + ca)

→ 2 > 3λ - 2 → 4/3 > λ

Therefore, λ < 4/3

Hope it's helpful to all