Explanation

(a)
Let $a^{\frac{1}{x}} = b^{\frac{1}{y}} = c^{\frac{1}{z}} = k$ [say]
$\Rightarrow \log a = x\log k,\ \log b = y\log k$
and $\log c = z\log k$
Since, $b^{2} = ac$
$\Rightarrow 2\log b = \log a + \log c$
$\Rightarrow 2\left( y\log k \right) = x\log k + z\log k$
$\Rightarrow 2y = x + z$
$\Rightarrow x,y,z$ are in AP.