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Properties of GP

If a, b, c ar...

Question

If a, b, c are real numbers such that $a + b + c = 0$ and $a^{2} + b^{2} + c^{2} = 1$ , then $(3 a + 5 b - 8 c)^{2} + (- 8 a + 3 b + 5 c)^{2} + (5 a - 8 b + 3 c)^{2}$ is equal to.

A

49

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B

98

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C

147

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D

294

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Solution

The correct option is A $147$
$(3 a + 5 b - 8 c)^{2} = 9 a^{2} + 25 b^{2} + 64 c^{2} + 30 a b - 80 b c - 48 a c$

$(- 8 a + 3 b + 5 c)^{2} = 64 a^{2} + 9 b^{2} + 25 c^{2} - 48 a b + 30 b c + - 80 a c$

$(5 a - 8 b + 3 c)^{2} = 25 a^{2} + 64 b^{2} + 9 c^{2} - 80 a b - 48 b c + 30 a c$

adding all three

$= 9 + 25 + 64 + 30 (a b + b c + c a) - 80 (a b + b c + c a) - 48 (a b + b c + c a)$

$= 98 - 98 (a b + b c + c a)$

$= 98 - 98 (((a + b + c)^{2} - a^{2} - b^{2} - c^{2}) / 2)$

$= 98 + 49$

$= 147$

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