If x and y are positive integer satisfying `tan^(-1)((1)/(x))+tan^(-1)((1)/(y))=(1)/(7)`, then the number of ordered pairs of (x,y) is

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the equation \( \tan^{-1}\left(\frac{1}{x}\right) + \tan^{-1}\left(\frac{1}{y}\right) = \frac{1}{7} \), we can use the formula for the sum of inverse tangents:

\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \]

provided \( ab < 1 \).

### Step 1: Apply the formula
Let \( a = \frac{1}{x} \) and \( b = \frac{1}{y} \). Then we can rewrite the equation as:

\[ \tan^{-1}\left(\frac{\frac{1}{x} + \frac{1}{y}}{1 - \frac{1}{xy}}\right) = \frac{1}{7} \]

### Step 2: Simplify the left-hand side
The left-hand side simplifies to:

\[ \tan^{-1}\left(\frac{\frac{y + x}{xy}}{1 - \frac{1}{xy}}\right) = \tan^{-1}\left(\frac{x + y}{xy - 1}\right) \]

### Step 3: Set the arguments equal
Since the inverse tangent function is one-to-one, we can set the arguments equal:

\[ \frac{x + y}{xy - 1} = \tan\left(\frac{1}{7}\right) \]

### Step 4: Cross-multiply
Cross-multiplying gives us:

\[ x + y = \tan\left(\frac{1}{7}\right)(xy - 1) \]

### Step 5: Rearranging the equation
Rearranging this equation leads to:

\[ xy \tan\left(\frac{1}{7}\right) - x - y = \tan\left(\frac{1}{7}\right) \]

### Step 6: Substitute \( k = \tan\left(\frac{1}{7}\right) \)
Let \( k = \tan\left(\frac{1}{7}\right) \). The equation becomes:

\[ xy k - x - y = k \]

### Step 7: Rearranging further
Rearranging gives:

\[ xy k - x - y - k = 0 \]

### Step 8: Factor the equation
This can be factored as:

\[ xy k - x - y + 1 - 1 - k = 0 \]

### Step 9: Solve for \( x \) in terms of \( y \)
From the equation, we can express \( x \):

\[ x = \frac{7y + 1}{y - 7} \]

### Step 10: Find integer solutions
Now, we need to find positive integer solutions for \( y \) such that \( y - 7 \) divides \( 7y + 1 \).

### Step 11: Check possible values for \( y \)
By checking integer values of \( y \) starting from 8 upwards (since \( y \) must be greater than 7):

- For \( y = 8 \): \( x = \frac{57}{1} = 57 \)
- For \( y = 9 \): \( x = \frac{64}{2} = 32 \)
- For \( y = 10 \): \( x = \frac{71}{3} \) (not an integer)
- For \( y = 11 \): \( x = \frac{78}{4} = 19.5 \) (not an integer)
- For \( y = 12 \): \( x = \frac{85}{5} = 17 \)
- For \( y = 13 \): \( x = \frac{92}{6} = 15.33 \) (not an integer)
- For \( y = 14 \): \( x = \frac{99}{7} = 14.14 \) (not an integer)
- For \( y = 15 \): \( x = \frac{106}{8} = 13.25 \) (not an integer)
- For \( y = 16 \): \( x = \frac{113}{9} = 12.56 \) (not an integer)
- For \( y = 17 \): \( x = \frac{120}{10} = 12 \)
- For \( y = 18 \): \( x = \frac{127}{11} = 11.545 \) (not an integer)
- For \( y = 19 \): \( x = \frac{134}{12} = 11.166 \) (not an integer)
- For \( y = 20 \): \( x = \frac{141}{13} = 10.846 \) (not an integer)
- For \( y = 21 \): \( x = \frac{148}{14} = 10.571 \) (not an integer)
- For \( y = 22 \): \( x = \frac{155}{15} = 10.333 \) (not an integer)
- For \( y = 23 \): \( x = \frac{162}{16} = 10.125 \) (not an integer)
- For \( y = 24 \): \( x = \frac{169}{17} = 9.941 \) (not an integer)
- For \( y = 25 \): \( x = \frac{176}{18} = 9.778 \) (not an integer)
- For \( y = 26 \): \( x = \frac{183}{19} = 9.632 \) (not an integer)
- For \( y = 27 \): \( x = \frac{190}{20} = 9.5 \) (not an integer)
- For \( y = 28 \): \( x = \frac{197}{21} = 9.333 \) (not an integer)
- For \( y = 29 \): \( x = \frac{204}{22} = 9.273 \) (not an integer)
- For \( y = 30 \): \( x = \frac{211}{23} = 9.174 \) (not an integer)
- For \( y = 31 \): \( x = \frac{218}{24} = 9.083 \) (not an integer)
- For \( y = 32 \): \( x = \frac{225}{25} = 9 \)

Continuing this process, we find valid pairs.

### Final Step: Count valid pairs
The valid pairs found are:

1. \( (57, 8) \)
2. \( (32, 9) \)
3. \( (17, 12) \)
4. \( (12, 17) \)
5. \( (9, 32) \)
6. \( (8, 57) \)

Thus, the total number of ordered pairs \( (x, y) \) is **6**.