Question

If $x,y,z$ are in AP, then $\frac{1}{\sqrt{x}+\sqrt{y}},\frac{1}{\sqrt{z}+\sqrt{x}},$ $\frac{1}{\sqrt{y}+\sqrt{z}}$ are in:

• A. AP
• B. GP
• C. HP
• D. AP and HP
• E. AP and GP

Answer 1

Answer: (A)

$\because$ $\frac{1}{\sqrt{z}+\sqrt{x}}-\frac{1}{\sqrt{x}+\sqrt{y}}$
$=\frac{1}{\sqrt{y}+\sqrt{z}}-\frac{1}{\sqrt{z}+\sqrt{x}}$
$\Rightarrow$ $y-z=x-y$
$\Rightarrow$ $y=\frac{z+x}{2}$
$\Rightarrow$ $x,y,z$ are in AP Hence,
$\frac{1}{\sqrt{x}+\sqrt{y}},\frac{1}{\sqrt{z}+\sqrt{x}},\frac{1}{\sqrt{y}+\sqrt{z}}$ are in AP.