For `0ltx ,yltpi,`the number of ordered pairs `(x, y)` satisfying system equations `cot^2(x-y)-(1+sqrt(3))cot(x-y)+sqrt(3)="0 and cosy="(sqrt(3))/2`is

Options

A. 0
B. 1
C. 2
D. 3

Answer

C

Explanation

We have `cos y=sqrt(3)/2`
`rArr y=pi/6`
Let `cot (x-y)=t`
`:. t^(2)-t-sqrt(3) t+sqrt(3)=0`
`rArr (t-1) (t-sqrt(3))=0`
`:. cot (x-y)=1 or cot (x-y)=sqrt(3)`
Given that `x in (0, pi)`
If `x-y=pi/6` then `x=pi/6+pi/6=pi/3 rArr (x, y) =(pi/3, pi/6)`
If `x-y=pi/4` then `x=pi/6+pi/4=(5pi)/12 rArr (x, y) = ((5pi)/12, pi/6)`
Possible ordered pairs (x, y) are `(pi/3, pi/6), ((5pi)/12, pi/6)`.