If $a, b, c, d$ are in continued proportion, then Whether the given equation $(b + c) (b + d) = (c + b) (c + d)$ is ?

True

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False

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Solution

The correct option is B False

Given $a, b, c, d$ are in continued proportion

$⟹ \frac{a}{b} = \frac{b}{c} = \frac{c}{d} = k$ (say)

$⟹ c = d k, b = c k = k^{2} d, a = b k = k^{3} d$

$∴ (a + d) (b + c) - (a + c) (b + d) = (k^{3} d + d) (k^{2} d + k d) - (k^{3} d + k d) (k^{2} d + d$
$= d^{2} (k^{5} + k^{2} + k^{4} + k - k^{5} - 2 k^{3} - k)$
$= k^{2} d^{2} (k - 1)^{2} = (k^{2} d - k d)^{2} = (b - c)^{2}$

Hence $(a + d) (b + c) \neq (a + c) (b + d)$

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