Question

For some $\alpha ϵR,$ if $\frac{\alpha -x}{px}=\frac{\alpha -y}{qy}=\frac{\alpha -z}{rz}$ and p,q,r are in A.P., then

• A. x, y, z are in H.P.
• B. x, y, z are in G.P.
• C. $x^2,y^2,z^2$ are in A.P.
• D. No such relation in x, y, z

Answer 1

The correct option is A

x, y, z are in H.P.

Let $\frac{\alpha -x}{px}=\frac{\alpha -y}{qy}=\frac{\alpha -z}{rz}=\lambda$

$\Rightarrow p=\frac{\alpha -x}{\lambda x},q=\frac{\alpha -y}{\lambda y},r=\frac{\alpha -z}{\lambda z}$

Given, $2q=p+r\Rightarrow \frac{2\alpha }{\lambda y}-\frac{2}{\lambda }=\frac{\alpha }{\lambda x}-\frac{1}{\lambda }+\frac{\alpha }{\lambda z}-\frac{1}{\lambda }$

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