The number of ordered pairs `(x,y)` , where `x`, `y in N` for which `4`, `x`, `y` are in `H.P.` , is equal to

Options

A. `1`
B. `2`
C. `3`
D. `4`

Explanation

To find the number of ordered pairs \((x, y)\) where \(x, y \in \mathbb{N}\) such that \(4, x, y\) are in Harmonic Progression (H.P.), we can follow these steps:

### Step 1: Understanding Harmonic Progression
For three numbers \(a\), \(b\), and \(c\) to be in H.P., their reciprocals \(\frac{1}{a}\), \(\frac{1}{b}\), and \(\frac{1}{c}\) must be in Arithmetic Progression (A.P.). Thus, we have:
\[ \frac{1}{4}, \frac{1}{x}, \frac{1}{y} \text{ are in A.P.} \]

### Step 2: Setting up the A.P. condition
From the property of A.P., we know:
\[ 2 \cdot \frac{1}{x} = \frac{1}{4} + \frac{1}{y} \]
This can be rearranged to:
\[ \frac{2}{x} = \frac{1}{4} + \frac{1}{y} \]

### Step 3: Finding a common denominator
To simplify, we find a common denominator:
\[ \frac{2}{x} = \frac{y + 4}{4y} \]
Cross-multiplying gives:
\[ 2 \cdot 4y = x(y + 4) \]
This simplifies to:
\[ 8y = xy + 4x \]

### Step 4: Rearranging the equation
Rearranging the equation, we have:
\[ xy + 4x - 8y = 0 \]
This can be expressed as:
\[ xy - 8y + 4x = 0 \]
Factoring out \(y\):
\[ y(x - 8) = -4x \]
Thus,
\[ y = \frac{4x}{8 - x} \]

### Step 5: Conditions for \(x\) and \(y\)
Since \(x\) and \(y\) must be natural numbers, the denominator \(8 - x\) must be a positive integer. Therefore, we have:
\[ 8 - x > 0 \implies x < 8 \]
Also, \(8 - x\) must be a factor of \(32\) for \(y\) to be a natural number.

### Step 6: Finding factors of \(32\)
The factors of \(32\) are \(1, 2, 4, 8, 16, 32\). We need to consider the values of \(8 - x\) that are factors of \(32\) and also satisfy \(x < 8\):
- If \(8 - x = 1 \Rightarrow x = 7\)
- If \(8 - x = 2 \Rightarrow x = 6\)
- If \(8 - x = 4 \Rightarrow x = 4\)
- If \(8 - x = 8 \Rightarrow x = 0\) (not valid as \(x\) must be natural)
- If \(8 - x = 16\) or \(32\) (not valid as \(x\) would be negative)

### Step 7: Finding corresponding \(y\) values
Now we calculate \(y\) for valid \(x\) values:
1. For \(x = 7\):
\[ y = \frac{4 \cdot 7}{8 - 7} = \frac{28}{1} = 28 \]
Ordered pair: \((7, 28)\)

2. For \(x = 6\):
\[ y = \frac{4 \cdot 6}{8 - 6} = \frac{24}{2} = 12 \]
Ordered pair: \((6, 12)\)

3. For \(x = 4\):
\[ y = \frac{4 \cdot 4}{8 - 4} = \frac{16}{4} = 4 \]
Ordered pair: \((4, 4)\)

### Step 8: Conclusion
The valid ordered pairs are \((7, 28)\), \((6, 12)\), and \((4, 4)\). Thus, the total number of ordered pairs \((x, y)\) is:
\[ \boxed{3} \]