(b) $y – x,\;2(y – a),\;(y – z)$ are in $H.P.$

$ \Rightarrow $ $\frac{1}{{y – x}},\;\frac{1}{{2(y – a)}},\;\frac{1}{{y – z}}$are in A.P.

$ \Rightarrow $ $\frac{1}{{2(y – a)}} – \frac{1}{{y – x}} = \frac{1}{{y – z}} – \frac{1}{{2(y – a)}}$

$ \Rightarrow $ $\frac{{2a – y – x}}{{y – x}} = \frac{{y + z – 2a}}{{y – z}}$

$ \Rightarrow $ $\frac{{(x – a)+ (y – a)}}{{(x – a) – (y -a)}} = \frac{{(y – a) + (z – a)}}{{(y – a) – (z – a)}}$

$ \Rightarrow $ $\frac{{x – a}}{{y – a}} = \frac{{y – a}}{{z – a}}$

(Applying componendo and dividendo)

$ \Rightarrow $ $x – a,\;y – a,\;z – a$ are in $G.P.$