If $a \neq b, a^{3} - b^{3} = 19 x^{3}$ and $a - b = x$ , then which of the following conclusions is correct?

a = 3 x

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a = 3 x

a = - 2 x

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a = - 3 x

a = 2 x

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a = 3 x

a = 2 x

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Solution

The correct option is A $a = 3 x$ or $a = - 2 x$
$a - b^{3} = (a - b) (a^{2} + a b + b^{2})$ or $x (a^{2} + a b + b^{2}) = 19 x^{3}$
Dividing by x and substituting $b = a - x$ , we obtain
$18 x^{2} + 3 a x - 3 a^{2} = 0$ or $6 x^{2} + a x - a^{2} = 0$
or $(3 x - a) (2 x + a) = 0$
i.e., $a = 3 x$ or $a = - 2 x$

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