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

The correct option is C H.P.

$\Rightarrow$ Let $\frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz}=k$

$\therefore$ $p=\frac{a-x}{xk},$ $q=\frac{a-y}{yk}$ and $r=\frac{a-z}{zk}$

It is given that, $p,q,r$ are in $A.P.$

$\therefore$ $2q=p+r$

$\Rightarrow$ $2\times \left(\frac{a-y}{yk}\right)=\frac{a-x}{xk}+\frac{a-z}{zk}$

$\Rightarrow$ $2\times \left(\frac{a-y}{y}\right)=\left(\frac{a-x}{x}\right)+\left(\frac{a-z}{z}\right)$

$\Rightarrow$ $2\times (\frac{a}{y}-1)=\frac{a}{x}-1+\frac{a}{z}-1$

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

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

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

$\therefore$ $x,y,z$ are in $H.P.$