Question

If a, b, c are three numbers such that ab + bc + ca = 0 and the average of a,b,c is x. Then prove that the average
of a2, b2 and c2 is 3x2.
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Answer 1

Required Answer:-

Given:

• a, b and c are three numbers.
• ab + bc + ac = 0
• Average of a, b and c is x.

To prove:

• Average of a², b² and c² is 3x²

Proof:

Given,

➡ ab + bc + ac = 0

➡ Average of a, b, c = x

➡ (a + b + c)/3 = x

➡ 3x = a + b + c

Now, we know that,

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

Putting the values here, we get,

➡ (3x)² = a² + b² + c² + 2 × 0

➡ 9x² = a² + b² + c²

Dividing both sides by 3, we get,

➡ (a² + b² + c²)/3 = 9x²/3

➡ Average of a², b² and c² = 3x² (Hence Proved)