Question

Suppose a fixed real number such that $\frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz}$ if $p,q,r$ in $AP$ , then $x,y,z$ are in

• A. $AP$
• B. $GP$
• C. $HP$
• D. $AGP$

Answer 1

The correct option is C $HP$

$Let\frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz}=k\left(say\right)$
$p=\frac{a-x}{kx},q=\frac{a-y}{ky},r=\frac{a-z}{kz}$
since $p,q,r$ in $AP$. So
$2p=q+r\Rightarrow 2\times \frac{a-y}{ky}=\frac{a-x}{kx}+\frac{a-z}{kz}$
$\Rightarrow \frac{2a}{y}-2=ax-1+\frac{a}{z}-1$

$\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{y}$

So $x,y,z$ are in $H.P$