Question

If $\frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz}$ and $p,q,r$ be in A.P., then $x,y$, $z$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (C)

We have, $\frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz}=\lambda$ (say)
$\Rightarrow p=\frac{a-x}{\lambda x},q=\frac{a-y}{\lambda y},r=\frac{a-z}{\lambda z}$
Now $p,q,r$ are in $A.P$.
$\Rightarrow \frac{a-x}{\lambda x},\frac{a-y}{\lambda y},\frac{a-z}{\lambda z}$ are in $A.P$.
$\Rightarrow \frac{a-x}{x},\frac{a-y}{y},\frac{a-z}{z}$ are in $A.P$.
$\Rightarrow \frac{a}{x}-1,\frac{a}{y}-1,\frac{a}{z}-1$ are in $A.P$.
$\Rightarrow \frac{a}{x},\frac{a}{y},\frac{a}{z}$ are in $A.P.$
$\Rightarrow \frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in $A.P.$
$\Rightarrow x,y,z$ are in $HP.$