Latest Gate Aptitude Question SOLUTION: If a2 + b2 + c2 = 1, then ab + bc + ac lies in the interval

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(#M40153766) GATE Numerical Ability question maths Keep an EYE

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

Asked In Gate PIYUSH AGRAWAL (9 years ago)

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oman jangir  (9 years ago)

using: (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)

(a + b + c)^2 ≥ 0 , then:

a^2 + b^2 + c^2 + 2(ab + bc + ca) ≥ 0

1 + 2(ab + bc + ca) ≥ 0

2(ab + bc + ca) ≥ -1

ab + bc + ca ≥ -1/2

now using: (a - b)^2 + (b - c)^2 + (c - a)^2 = 2(a^2 + b^2 + c^2) - 2(ab + bc + ca)

(a - b)^2 + (b - c)^2 + (c - a)^2 ≥ 0

2(a^2 + b^2 + c^2) - 2(ab + bc + ca) ≥ 0

2(1) - 2(ab + bc + ca) ≥ 0

-2(ab + bc + ca) ≥ -2

ab + bc + ca ≤ 1

hence:

-1/2 ≤ ab + bc + ca ≤ 1

[-1/2 , 1]

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