The number of positive integral solutions of ` tan^(-1)x + cot^(-1)y= tan^(-1)3 ` is :

Options

A. one
B. two
C. three
D. four

Answer

To solve the equation \( \tan^{-1} x + \cot^{-1} y = \tan^{-1} 3 \) for positive integral solutions, we can follow these steps:

### Step 1: Use the identity for cotangent
We know that \( \cot^{-1} y = \frac{\pi}{2} - \tan^{-1} y \). Therefore, we can rewrite the equation as:
\[ \tan^{-1} x + \left( \frac{\pi}{2} - \tan^{-1} y \right) = \tan^{-1} 3 \]
This simplifies to:
\[ \tan^{-1} x - \tan^{-1} y = \tan^{-1} 3 - \frac{\pi}{2} \]

### Step 2: Rearranging the equation
Rearranging gives us:
\[ \tan^{-1} x - \tan^{-1} y = -\left(\frac{\pi}{2} - \tan^{-1} 3\right) \]
Using the property \( \tan^{-1} a - \tan^{-1} b = \tan^{-1} \left( \frac{a-b}{1+ab} \right) \), we can express this as:
\[ \tan^{-1} \left( \frac{x - y}{1 + xy} \right) = -\tan^{-1} \left( \frac{1}{3} \right) \]

### Step 3: Convert to tangent
Taking the tangent of both sides, we have:
\[ \frac{x - y}{1 + xy} = -\frac{1}{3} \]
Cross-multiplying gives:
\[ 3(x - y) = - (1 + xy) \]
Rearranging yields:
\[ 3x - 3y + xy + 1 = 0 \]

### Step 4: Rearranging to find y
Rearranging this equation gives:
\[ xy + 3x + 1 = 3y \]
This can be rewritten as:
\[ xy - 3y + 3x + 1 = 0 \]
Factoring out \( y \):
\[ y(x - 3) = - (3x + 1) \]
Thus, we have:
\[ y = \frac{- (3x + 1)}{x - 3} \]

### Step 5: Finding positive integral solutions
For \( y \) to be a positive integer, \( -(3x + 1) \) must be divisible by \( (x - 3) \).

1. **Case \( x = 1 \)**:
\[ y = \frac{-(3(1) + 1)}{1 - 3} = \frac{-4}{-2} = 2 \]
So, one solution is \( (1, 2) \).

2. **Case \( x = 2 \)**:
\[ y = \frac{-(3(2) + 1)}{2 - 3} = \frac{-7}{-1} = 7 \]
So, another solution is \( (2, 7) \).

3. **Case \( x = 3 \)**:
\[ y = \frac{-(3(3) + 1)}{3 - 3} \text{ (undefined)} \]
Thus, no solution for \( x = 3 \).

4. **Case \( x > 3 \)**:
As \( x \) increases, \( y \) becomes negative or undefined, leading to no further positive integral solutions.

### Conclusion
The only positive integral solutions are \( (1, 2) \) and \( (2, 7) \). Therefore, the number of positive integral solutions is:
\[ \boxed{2} \]