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

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Solution

If $a, b, c, d$ are in G.P, then the value of
Let $y = (a - c)^{2} + (b - c)^{2} + (b - d)^{2} - (a - d)^{2}$
$y = a^{2} - 2 a c + c^{2} + b^{2} - 2 b c + c^{2} + b^{2} - 2 b d + d^{2} - (a^{2} - 2 a d + d^{2})$
$y = 2 (b^{2} - a c) + 2 (c^{2} - b d) + 2 (a d - b c)$
As $a, b, c, d$ are in GP then $b^{2} - a c = 0, c^{2} - b d = 0$ and $a d = b c$
So $y = 0$

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