Quadratic Equation.

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

Expert Solution

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]

Nikhil Thakur answered 2 years ago

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