If `x, y, z` are distinct positive real numbers is A.P. then `(1)/(sqrt(x)+sqrt(y)), (1)/(sqrt(z)+sqrt(x)), (1)/(sqrt(y)+sqrt(z))` are in

Options

A. `A.P.`
B. `G.P.`
C. `H.P.`
D. `A.G.P.`

Explanation

To determine whether the numbers \(\frac{1}{\sqrt{x}+\sqrt{y}}, \frac{1}{\sqrt{z}+\sqrt{x}}, \frac{1}{\sqrt{y}+\sqrt{z}}\) are in arithmetic progression (A.P.), we can follow these steps:

### Step 1: Understand the condition for A.P.
Three numbers \(a\), \(b\), and \(c\) are in A.P. if:
\[ 2b = a + c \]

### Step 2: Assign the values
Let:
\[ a = \frac{1}{\sqrt{x}+\sqrt{y}}, \quad b = \frac{1}{\sqrt{z}+\sqrt{x}}, \quad c = \frac{1}{\sqrt{y}+\sqrt{z}} \]

### Step 3: Set up the A.P. condition
We need to check if:
\[ 2b = a + c \]
This translates to:
\[ 2 \cdot \frac{1}{\sqrt{z}+\sqrt{x}} = \frac{1}{\sqrt{x}+\sqrt{y}} + \frac{1}{\sqrt{y}+\sqrt{z}} \]

### Step 4: Find a common denominator
The right-hand side can be simplified by finding a common denominator:
\[ \frac{1}{\sqrt{x}+\sqrt{y}} + \frac{1}{\sqrt{y}+\sqrt{z}} = \frac{(\sqrt{y}+\sqrt{z}) + (\sqrt{x}+\sqrt{y})}{(\sqrt{x}+\sqrt{y})(\sqrt{y}+\sqrt{z})} \]
This simplifies to:
\[ \frac{\sqrt{x} + 2\sqrt{y} + \sqrt{z}}{(\sqrt{x}+\sqrt{y})(\sqrt{y}+\sqrt{z})} \]

### Step 5: Multiply through by the common denominator
To eliminate the fractions, we multiply both sides by \((\sqrt{x}+\sqrt{y})(\sqrt{y}+\sqrt{z})(\sqrt{z}+\sqrt{x})\):
\[ 2(\sqrt{x}+\sqrt{y})(\sqrt{y}+\sqrt{z}) = (\sqrt{y}+\sqrt{z})(\sqrt{z}+\sqrt{x}) + (\sqrt{x}+\sqrt{y})(\sqrt{y}+\sqrt{z}) \]

### Step 6: Expand and simplify
Expanding both sides gives:
- Left-hand side:
\[ 2(\sqrt{x}\sqrt{y} + \sqrt{y}\sqrt{z} + \sqrt{x}\sqrt{z} + \sqrt{y}\sqrt{z}) = 2(\sqrt{xy} + \sqrt{yz} + \sqrt{zx}) \]
- Right-hand side:
\[ (\sqrt{yz} + \sqrt{zx} + \sqrt{xy} + \sqrt{yz}) + (\sqrt{xy} + \sqrt{yz} + \sqrt{zx} + \sqrt{xy}) = 2(\sqrt{xy} + \sqrt{yz} + \sqrt{zx}) \]

### Step 7: Conclusion
Since both sides are equal, we conclude that:
\[ \frac{1}{\sqrt{x}+\sqrt{y}}, \frac{1}{\sqrt{z}+\sqrt{x}}, \frac{1}{\sqrt{y}+\sqrt{z}} \text{ are in A.P.} \]

### Final Answer
Thus, the given numbers are in A.P.

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