Let $\mathrm{x}^{2}, \mathrm{y}^{2}, \mathrm{z}^{2}$ are in A.P $\Rightarrow \mathrm{y}^{2}-\mathrm{x}^{2}=\mathrm{z}^{2}-\mathrm{y}^{2}$
$2 y^{2}=x^{2}+z^{2}$
(a) Suppose $\mathrm{y}+\mathrm{z}, \mathrm{z}+\mathrm{x}$ and $\mathrm{x}+\mathrm{y}$ are in $\mathrm{A}-\mathrm{P}$
$\therefore(\mathrm{z}+\mathrm{x})-(\mathrm{y}+\mathrm{z})=(\mathrm{x}+\mathrm{y})-(\mathrm{z}+\mathrm{x})$
$2(z+x)=(y+z)+(x+y)$
$\Rightarrow 2 z+2 x=2 y+z+x \Rightarrow z+x=2 y$
$\Rightarrow \mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ are in $\mathrm{AP}$. Which is true
(b) Let $\mathrm{y}+\mathrm{z}, \mathrm{z}+\mathrm{x}, \mathrm{x}+\mathrm{y}$ are in $\mathrm{HP}$.
$z+x=\frac{2(y+z)(x+y)}{y+z+x+y}$
$\Rightarrow \quad z+x=\frac{2(y+z)(x+y)}{2 y+z+x}$
$\Rightarrow \quad 2 y z+z^{2}+z x+2 x y+x z+x^{2}$
$\quad=2 y x+2 y^{2}+2 z x+2 y z$
$\Rightarrow z^{2}+x^{2}=2 y^{2}$
$\Rightarrow \mathrm{x}^{2}, \mathrm{y}^{2}$ and $\mathrm{z}^{2}$ are in AP. Which is true. Hence, $\mathrm{y}+\mathrm{z}, \mathrm{z}+\mathrm{x}$ and $\mathrm{x}+$ y are in A.P.