If $a, b, c$ be distinct real numbers such that $a^{2} - b = b^{2} - c = c^{2} - a$ , then $(a + b) (b + c) (c + a)$ , is equal to

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Solution

The correct option is B $1$
Given, $a^{2} - b = b^{2} - c = c^{2} - a$
$\Rightarrow a^{2} - b^{2} = b - c, b^{2} - c^{2} = c - a, c^{2} - a^{2} = a - b$
$\Rightarrow a + b = \frac{b - c}{a - b}, b + c = \frac{c - a}{b - c}, c + a = \frac{a - b}{c - a}$
$∴ (a + b) (b + c) (c + a) = 1$

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