Let 0 z y x be three real numbers such that 1 x , 1 y , 1 z are in an arithmetic progression and x , 2 y , z are in a geometric progression. If x y + y z + z x = 3 2 x y z , then 3 ( x + y + z ) 2 is equal to _________ . [2023]

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Solution

Q.
Let $0 < z < y < x$ be three real numbers such that $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in an arithmetic progression and $x, \sqrt{2} y, z$ are in a geometric progression. If $x y + y z + z x = \frac{3}{\sqrt{2}} x y z,$ then $3 {(x + y + z)}^{2}$ is equal to _________ . [2023]

Ans.
(150)

$\frac{2}{y} = \frac{1}{x} + \frac{1}{z} and 2 y^{2} = x z$

$\Rightarrow \frac{2}{y} = \frac{x + z}{x z} = \frac{x + z}{2 y^{2}} \Rightarrow x + z = 4 y$

Now, $x y + y z + z x = \frac{3}{\sqrt{2}} x y z \Rightarrow y (x + z) + z x = \frac{3}{\sqrt{2}} x z \cdot y$

$\Rightarrow 4 y^{2} + 2 y^{2} = \frac{3}{\sqrt{2}} y \cdot 2 y^{2} \Rightarrow 6 y^{2} = 3 \sqrt{2} y^{3} \Rightarrow y = \sqrt{2}$

$∴$ $x + y + z = 5 y = 5 \sqrt{2} \Rightarrow 3 {(x + y + z)}^{2} = 3 \times 50 = 150$

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