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`
For `lambda=11`, an integral pair `(m, n)` satisfying the equation is:

Options

A. (a) `(12,72)`
B. (b) `(12, 133)`
C. (c) `(13,74)`
D. (d) `(13,136)`

Answer

To solve the equation \( \tan^{-1}\left(\frac{1}{m}\right) + \tan^{-1}\left(\frac{1}{n}\right) = \tan^{-1}\left(\frac{1}{\lambda}\right) \) for integral pairs \( (m, n) \) when \( \lambda = 11 \), we will follow these steps:

### Step 1: Use the formula for the sum of inverse tangents
The formula for the sum of two inverse tangents is given by:
\[ \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \]
Applying this to our equation, we have:
\[ \tan^{-1}\left(\frac{1}{m}\right) + \tan^{-1}\left(\frac{1}{n}\right) = \tan^{-1}\left(\frac{\frac{1}{m} + \frac{1}{n}}{1 - \frac{1}{mn}}\right) \]

### Step 2: Simplify the left-hand side
The left-hand side becomes:
\[ \tan^{-1}\left(\frac{\frac{n+m}{mn}}{1 - \frac{1}{mn}}\right) = \tan^{-1}\left(\frac{n+m}{mn - 1}\right) \]
Thus, we can rewrite our equation as:
\[ \tan^{-1}\left(\frac{n+m}{mn - 1}\right) = \tan^{-1}\left(\frac{1}{\lambda}\right) \]

### Step 3: Equate the arguments of the inverse tangent
Since the inverse tangent function is one-to-one, we can equate the arguments:
\[ \frac{n+m}{mn - 1} = \frac{1}{\lambda} \]
Cross-multiplying gives:
\[ (n+m)\lambda = mn - 1 \]

### Step 4: Rearranging the equation
Rearranging the equation, we have:
\[ mn - \lambda n - \lambda m = -1 \]
This can be rewritten as:
\[ mn - \lambda n = \lambda m - 1 \]
Factoring out \( n \) from the left side:
\[ n(m - \lambda) = \lambda m + 1 \]

### Step 5: Solve for \( n \)
Now, we can solve for \( n \):
\[ n = \frac{\lambda m + 1}{m - \lambda} \]
Substituting \( \lambda = 11 \):
\[ n = \frac{11m + 1}{m - 11} \]

### Step 6: Find integral pairs \( (m, n) \)
We need \( n \) to be an integer, which means \( m - 11 \) must divide \( 11m + 1 \). We can check various values of \( m \):

1. **For \( m = 12 \)**:
\[ n = \frac{11 \times 12 + 1}{12 - 11} = \frac{132 + 1}{1} = 133 \]
So, one pair is \( (12, 133) \).

2. **For \( m = 13 \)**:
\[ n = \frac{11 \times 13 + 1}{13 - 11} = \frac{143 + 1}{2} = \frac{144}{2} = 72 \]
So, another pair is \( (13, 72) \).

### Conclusion
The integral pairs \( (m, n) \) that satisfy the equation for \( \lambda = 11 \) are \( (12, 133) \) and \( (13, 72) \). Among the given options, the correct integral pair is \( (12, 133) \).