cat 2023 Question 20

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

Accepted Answer

Given that $ \frac{1}{\sqrt{y} + \sqrt{z}} $ is the arithmetic mean of $ \frac{1}{\sqrt{x} + \sqrt{z}} $ and $ \frac{1}{\sqrt{x} + \sqrt{y}} $

=> $ \frac{2}{\sqrt{y} + \sqrt{z}} = \frac{1}{\sqrt{x} + \sqrt{z}} + \frac{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.

Question 20.

Question Explanation