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If x,y,z are real and $4 x^{2} + 9 y^{2} + 16 z^{2} - 6 x y - 12 y z - 8 z x = 0$ then x,y,z are

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Detailed Solution

Multiply the given expression by 2 and write it as

$(2 x - 3 y)^{2} + (3 y - 4 z)^{2} + (4 z - 2 x)^{2} = 0$

$\Rightarrow 2 x = 3 y = 4 z \Rightarrow \frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in A.P.

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