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Geometric Progression

If a, b, c,...

Question

If $a, b, c, d$ are in G.P., then ${(a - c)}^{2} + {(c - b)}^{2} + {(b - d)}^{2} - {(d - a)}^{2}$ is

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-1

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None of these

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Solution

The correct option is C 0
Given expression $= a^{2} + c^{2} - 2 a c + c^{2} + b^{2} - 2 b c + b^{2} + d^{2} - 2 b d - d^{2} - a^{2} + 2 a d$

$= a^{2} + c^{2} - 2 b^{2} + c^{2} + b^{2} - 2 b c + b^{2} + d^{2} - 2 c^{2} - d^{2} - a^{2} + 2 a d$

$= 2 a d - 2 b c$ $[∵ a d = b c$ they are in GP]

$= 0$

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