Question

If a² + b² + c² = 1, then ab + bc + ac lies in which interval

Answer 1

Given that a² + b² + c² = 1 …(i)

 

We know (a+b+c)² ≥ 0

 

a²+b²+c²+2ab+2bc+2ac ≥ 0

 

=> 1 + 2(ab+bc+ac)≥ 0 (from (i))

 

=> 2(ab+bc+ac)≥ -1

 

=> (ab+bc+ac)≥ -½ …(ii)

 

We know that ½ [(a-b)² + (b-c)² + (c-a)²]≥ 0

 

=> a² + b² + c² – ab – bc – ac ≥ 0

 

=> ab + bc + ac ≤ 1 …(iii)

 

Combining (ii) and (iii)

 

-½ ≤ (ab+bc+ac) ≤ 1

 

Therefore (ab+bc+ac) ∈ [-½, 1]

 

 

 (ab+bc+ac) ∈ [-½, 1]