CAT 2023 Slot 1 QA Question & Solution
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
Options
Solution
Given that $\dfrac{1}{\sqrt{y}+\sqrt{z}}$ is the arithmetic mean of $\dfrac{1}{\sqrt{x}+\sqrt{z}}$ and $\dfrac{1}{\sqrt{x}+\sqrt{y}}$
=> $\dfrac{2}{\sqrt{\ y}+\sqrt{\ z}}=\dfrac{1}{\sqrt{\ x}+\sqrt{\ z}}+\dfrac{1}{\sqrt{\ x}+\sqrt{\ y}}$
=> $2\left(\sqrt{\ x}+\sqrt{\ z}\right)\left(\sqrt{\ x}+\sqrt{\ y}\right)=\left(\sqrt{\ y}+\sqrt{\ z}\right)\left(\sqrt{\ x}+\sqrt{\ y}+\sqrt{\ x}+\sqrt{\ z}\right)$
=> $2\left(x+\sqrt{\ xy}+\sqrt{\ xz}+\sqrt{\ yz}\right)=2\sqrt{xy}+y+\sqrt{\ yz}+2\sqrt{\ xz}+\sqrt{\ yz}+z$
=> $2x=y+z$
=> y, x, z are in A.P. as x is the arithmetic mean of y and z.