Question

If $\frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{r}$ and $p,q,r$ are in AP then prove that $x,y,z$ are in HP.

Answer 1

Given $p,q,r$ are in $AP$ $⟹2q=p+r$

Given $\frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{r}=k\left(say\right)$

$⟹p=\frac{a}{kx}-\frac{1}{k}$

Similarly $q=\frac{a}{ky}-\frac{1}{k}$ $,r=\frac{a}{kz}-\frac{1}{k}$

$2q=p+r$

$⟹\frac{2a}{ky}-\frac{2}{k}=\frac{a}{kx}-\frac{1}{k}+\frac{a}{kz}-\frac{1}{k}$

$⟹\frac{a}{k}(\frac{2}{y})=\frac{a}{k}(\frac{1}{x}+\frac{1}{z})$

$⟹\frac{2}{y}=\frac{1}{x}+\frac{1}{z}$

$⟹x,y,z$ are in $HP$