If $a^{x} = b^{y} = c^{z}$ , $a b c = 1$ , then find the value of $x y + y z + z x$ .

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Solution

Let $a^{x} = b^{y} = c^{z} = k$

$∴ a = k^{1 / x}, b = k^{1 / y}, c = k^{1 / z}$

Since,

$a b c = 1$

Therefore,

$k^{1 / x} . k^{1 / y} . k^{1 / z} = 1$

$k^{(1 / x + 1 / y + 1 / z)} = k^{0}$

$∴ 1 / x + 1 / y + 1 / z = 0$

$\frac{x y + y z + z x}{x y z} = 0$

$x y + y z + z x = 0$

Hence, this is the answer.

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