Explanation

(a)
Let $\frac{1}{\sqrt{x} + \sqrt{y}},\frac{1}{\sqrt{z} + \sqrt{x}},\frac{1}{\sqrt{y} + \sqrt{z}}$ are in AP.
Then, $\frac{1}{\sqrt{z} + \sqrt{x}} - \frac{1}{\sqrt{x} + \sqrt{y}} = \frac{1}{\sqrt{y} + \sqrt{z}} - \frac{1}{\sqrt{z} + \sqrt{x}}$
$\Rightarrow y - z = x - y\ \Rightarrow \ y = \frac{z + x}{2}$
$\Rightarrow x,y,z$ 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.