CAT 2023 Slot 1 QA Question 21: For some positive and distinct real numbers x, y and z, if $\dfrac{1}{\sqrt{y}+\sqrt{z}}$ is the arithmetic mean of $\dfrac{1}{\sqrt{x}+\sqrt{z}}$ and $\dfrac{1}{\sqrt{x}+\sqrt{y}}$, then the relation

Question

For some positive and distinct real numbers x, y and z, if $\dfrac{1}{\sqrt{y}+\sqrt{z}}$ is the arithmetic mean of $\dfrac{1}{\sqrt{x}+\sqrt{z}}$ and $\dfrac{1}{\sqrt{x}+\sqrt{y}}$, then the relation

Accepted Answer

Let $a=\sqrt{x}$, $b=\sqrt{y}$ and $c=\sqrt{z}$.

The given condition becomes
$\dfrac{1}{b+c} = \dfrac{1}{2} \left(\dfrac{1}{a+c} + \dfrac{1}{a+b} \right)$

Multiplying by 2,
$\dfrac{2}{b+c} = \dfrac{1}{a+c} + \dfrac{1}{a+b}$
$\dfrac{2}{b+c} = \dfrac{2a+b+c}{(a+c)(a+b)}$

Cross-multiplying,
$2(a+c)(

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

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