Discussion

Question:

For some positive and distinct real numbers x, y and z if $\frac{1}{\sqrt{y} + \sqrt{z}}$ is the arithmetic mean of $\frac{1}{\sqrt{x} + \sqrt{z}}$ and $\frac{1}{\sqrt{x} + \sqrt{y}}$ , then the relationship which will always hold true, is?

Explanation:

Given, $\frac{2}{\sqrt{y} + \sqrt{z}}$ = $\frac{1}{\sqrt{x} + \sqrt{z}}$ + $\frac{1}{\sqrt{x} + \sqrt{y}}$

⇒ $\frac{2}{\sqrt{y} + \sqrt{z}}$ = $\frac{\sqrt{x} + \sqrt{y} + \sqrt{x} + \sqrt{z}}{(\sqrt{x} + \sqrt{z}) (\sqrt{x} + \sqrt{y})}$

⇒ 2(√x + √z)(√x + √y) = (2√x + √y + √z)(√y + √z)

⇒ 2x + 2√xy + 2√zx + 2√zy = 2√xy + 2√xz + y + √yz + zy + z

⇒ 2x = y + z

∴ y, x and z are in Arithmetic Progression

Hence, option (b).

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