If `x , y , z` are natural numbers such that `cot^(-1)x+cot^(-1)y=cot^(-1)z` then the number of ordered triplets `(x , y , z)` that satisfy the equation is 0 (b) 1 (c) 2 (d) Infinite solutions

Options

A. 0
B. 1
C. 2
D. Infinite solution

Answer

D

Explanation

`x = 1, y =1` is not a solution of the given equation.
Suppose `(x, y) != (1,1)`
Then `((1)/(x) + (1)/(y))/(1-(1)/(xy)) = (1)/(z)`
`rArr x(z -y) = -(1 + yz)`
`rArr x = -((1 + xyz))/(z -y)`
If `y = n + 1, z = n " then " x = n^(2) + n+1`
All such numbers are solutions of the given equation