Question

If $\frac{x+y}{1-xy},y,\frac{y+z}{1-yz}$ are in $A.P.,$ then $x,\frac{1}{y},z$ will be in

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

Answer 1

The correct option is C H.P.

Since $\frac{x+y}{1-xy},y,\frac{y+z}{1-yz}$ are in $A.P.$

$\Rightarrow y-\frac{x+y}{1-xy}=\frac{y+z}{1-yz}-y\Rightarrow \frac{y-x{y}^2-x-y}{1-xy}=\frac{y+z-y+y^{2}z}{1-yz}$

$\Rightarrow \frac{-x}{1-xy}=\frac{z}{1-yz}\Rightarrow -x+xyz=z-xyz$

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

$\Rightarrow x,\frac{1}{y},z$ are in $H.P.$