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  4. If x,y,z are in AP, then (1/√x+√y),(1/√z+√x), (1/√y+√z) are in:

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Q. If $ x,y,z $ are in AP, then $ \frac{1}{\sqrt{x}+\sqrt{y}},\frac{1}{\sqrt{z}+\sqrt{x}}, $ $ \frac{1}{\sqrt{y}+\sqrt{z}} $ are in:

KEAMKEAM 2005

Solution:

$ \because $ $ \frac{1}{\sqrt{z}+\sqrt{x}}-\frac{1}{\sqrt{x}+\sqrt{y}} $
$ =\frac{1}{\sqrt{y}+\sqrt{z}}-\frac{1}{\sqrt{z}+\sqrt{x}} $
$ \Rightarrow $ $ y-z=x-y $
$ \Rightarrow $ $ y=\frac{z+x}{2} $
$ \Rightarrow $ $ x,\text{ }y,\text{ }z $ are in AP Hence,
$ \frac{1}{\sqrt{x}+\sqrt{y}},\frac{1}{\sqrt{z}+\sqrt{x}},\frac{1}{\sqrt{y}+\sqrt{z}} $ are in AP.