Sometimes we are just concerned with finding integral solutions to equations. Consider the equation
`tan^(-1).(1)/m+tan^(-1).(1)/n=tan^(-1).(1)/lambda`, where `m,n, lambda in N`
How many positive integral solutions `(m,n)` exist for the equation if `lambda=3`?

Options

A. Two
B. Four
C. Six
D. Eight

Answer

To solve the equation \( \tan^{-1}\left(\frac{1}{m}\right) + \tan^{-1}\left(\frac{1}{n}\right) = \tan^{-1}\left(\frac{1}{\lambda}\right) \) where \( \lambda = 3 \), we will follow these steps:

### Step 1: Use the formula for the sum of inverse tangents
We know that:
\[ \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \]
Applying this to our equation:
\[ \tan^{-1}\left(\frac{1}{m}\right) + \tan^{-1}\left(\frac{1}{n}\right) = \tan^{-1}\left(\frac{\frac{1}{m} + \frac{1}{n}}{1 - \frac{1}{mn}}\right) \]
This simplifies to:
\[ \tan^{-1}\left(\frac{\frac{n+m}{mn}}{\frac{mn-1}{mn}}\right) = \tan^{-1}\left(\frac{m+n}{mn-1}\right) \]

### Step 2: Set the right-hand side equal to the left-hand side
Now, we set this equal to the right-hand side:
\[ \tan^{-1}\left(\frac{m+n}{mn-1}\right) = \tan^{-1}\left(\frac{1}{3}\right) \]
This implies:
\[ \frac{m+n}{mn-1} = \frac{1}{3} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3(m+n) = mn - 1 \]
Rearranging this equation results in:
\[ mn - 3m - 3n = 1 \]

### Step 4: Rearranging the equation
We can rearrange this into a standard quadratic form:
\[ mn - 3m - 3n - 1 = 0 \]

### Step 5: Solve for \( m \) in terms of \( n \)
We can express \( m \) in terms of \( n \):
\[ m = \frac{3n + 1}{n - 3} \]
This gives us a formula for \( m \) based on the value of \( n \).

### Step 6: Finding positive integral solutions
To find positive integral solutions, we need \( n - 3 \) to be a divisor of \( 3n + 1 \).

Let \( k = n - 3 \), then \( n = k + 3 \). Substituting this into the equation:
\[ m = \frac{3(k + 3) + 1}{k} = \frac{3k + 9 + 1}{k} = \frac{3k + 10}{k} = 3 + \frac{10}{k} \]
For \( m \) to be an integer, \( k \) must divide 10. The positive divisors of 10 are 1, 2, 5, and 10.

### Step 7: Calculate corresponding \( n \) and \( m \)
1. If \( k = 1 \), then \( n = 4 \) and \( m = 13 \).
2. If \( k = 2 \), then \( n = 5 \) and \( m = 8 \).
3. If \( k = 5 \), then \( n = 8 \) and \( m = 5 \).
4. If \( k = 10 \), then \( n = 13 \) and \( m = 4 \).

### Step 8: Count the solutions
The positive integral solutions \( (m, n) \) are:
1. \( (13, 4) \)
2. \( (8, 5) \)
3. \( (5, 8) \)
4. \( (4, 13) \)

Thus, there are **4 positive integral solutions**.

### Final Answer
The number of positive integral solutions \( (m, n) \) is **4**.

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