Question

If (a + b + c) = 8√3, a² + b² + c² = 64, then find the ratio of a : b : c
1. 1 : 1 : 1
2. 1 : √2 : √3
3. √5 : √7 : √8
4. √7 : √8 : √5

Answer 1

Correct Answer - Option 1 : 1 : 1 : 1

Given

a + b + c = 8√3

a² + b² + c² = 64

Concept

a² + b² + c² = ab + bc + ca

Then, a = b = c

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

Calculation

(8√3)² = 64 + 2(ab + bc + ca)

⇒ 192 - 64 = 2(ab + bc + ca)

⇒ 128/2 = ab + bc + ca

⇒ 64 = ab + bc + ca

As a² + b² + c² = ab + bc + ca

So, a = b = c

b × b = ca

⇒ b² = ca

Similarly, ab = b² and bc = b²

64 = b² + b² + b²

64 = 3b²

64/3 = b²

8/√3 = b

a = b = c = 8/√3

⇒ 1 : 1 : 1

∴ a ∶ b ∶ c is 1 ∶ 1 ∶ 1