Question

If $x,y\mathrm{and}z$ 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

Answer 1

The correct option is A

AP

Explanation for the correct option:

Step 1: Express as per the condition of AP:

It is given that $x,y\mathrm{and}z$ in AP, so

$y-x=z-y.....\left(1\right)$

By taking reciprocal of $\left(1\right)$, we get

$$\begin{gathered} \frac{1}{y-x}=\frac{1}{z-y} \\ \Rightarrow \frac{1}{x-y}=\frac{1}{y-z}.......\left(2\right) \end{gathered}$$

Step 2: Use the algebraic identity ${\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}\mathbf{=}\left(a+b\right)\left(a-b\right)$:

We can write $\left(2\right)$ as

$\begin{array}{rcl}\frac{1}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}& =& \frac{1}{\left(\sqrt{y}+\sqrt{z}\right)\left(\sqrt{y}-\sqrt{z}\right)}\\ \Rightarrow \frac{\left(\sqrt{y}-\sqrt{z}\right)}{\left(\sqrt{x}+\sqrt{y}\right)}& =& \frac{\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{y}+\sqrt{z}\right)}\end{array}$

Add and subtract $\sqrt{x}$ on L.H.S and $\sqrt{y}$ on R.H.S:

$\begin{array}{rcl}\frac{\sqrt{x}+\sqrt{y}-\sqrt{z}-\sqrt{x}}{\sqrt{x}+\sqrt{y}}& =& \frac{\sqrt{z}+\sqrt{x}-\sqrt{y}-\sqrt{z}}{\sqrt{y}+\sqrt{z}}\\ & \Rightarrow & \frac{\left(\sqrt{x}+\sqrt{y}\right)-\left(\sqrt{z}+\sqrt{x}\right)}{\sqrt{x}+\sqrt{y}}=\frac{\left(\sqrt{z}+\sqrt{x}\right)-\left(\sqrt{y}+\sqrt{z}\right)}{\sqrt{y}+\sqrt{z}}\end{array}$

Divide both side by $\left(\sqrt{z}+\sqrt{x}\right)$:

$\begin{array}{rcl}\frac{\left(\sqrt{x}+\sqrt{y}\right)-\left(\sqrt{z}+\sqrt{x}\right)}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{z}+\sqrt{x}\right)}& =& \frac{\left(\sqrt{z}+\sqrt{x}\right)-\left(\sqrt{y}+\sqrt{z}\right)}{\left(\sqrt{y}+\sqrt{z}\right)\left(\sqrt{z}+\sqrt{x}\right)}\\ & \Rightarrow & \frac{1}{\left(\sqrt{z}+\sqrt{x}\right)}-\frac{1}{\left(\sqrt{x}+\sqrt{y}\right)}=\frac{1}{\left(\sqrt{y}+\sqrt{z}\right)}-\frac{1}{\left(\sqrt{z}+\sqrt{x}\right)}\end{array}$

Therefore, $\frac{1}{\sqrt{x}+\sqrt{y}},\frac{1}{\sqrt{z}+\sqrt{x}},\frac{1}{\sqrt{y}+\sqrt{z}}$ are in AP.

Hence, option A is correct.