Question

If a + b + c = 4√3 and a² + b² + c² = 16, then find a : b : c
1. 1 : 2 : 3
2. 3 : 2 : 1
3. 1 : 1 : 1
4. 1 : 2√3 : 1

Answer 1

Correct Answer - Option 3 : 1 : 1 : 1

Given-

a + b + c = 4√3

a2 + b2 + c2 = 16

Concept Used-

If ab + bc + ca = a2 + b2 + c2, then a = b = c

Formula Used-

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

Calculation-

a + b + c = 4√3

⇒ (a + b + c)2 = 48

⇒ a2 + b2 + c2 + 2ab + 2bc + 2ca = 48

⇒ 2(ab + bc + ca) = 48 - 16

⇒ ab + bc + ca = 16

We can see that a2 + b2 + c2 = ab + bc + ca = 16

This is only possible when a = b = c.

∴ a : b : c = 1 : 1 : 1