The number of solution of the equation `sqrt(13-18tanx)=6 tan x-3`, where `-2pilt x lt 2pi`is

Options

A. 0
B. 2
C. 4
D. 8

Answer

C

Explanation

`sqrt(13-18tan x)=6 tan x-3 -2pi lt x lt 2 pi` (given)
`rArr 13-18 tan x = 36 tan^(2)x + 9-36 tan x`
`rArr 36 tan^(2)x - 18 tan x-4 =0`
`rArr 36 tan^(2) x-18 tan x -4=0`
`rArr 36 tan^(2)x-24 tan x + 6 tan x-4 =0`
`rArr 6 tan x(6 tan x +1) -4(6 tan x+1)=0`
`tan x=(2)/(3)` or `tan x=-1//6` (not possible)
Let `tan alpha = 2//3`
then `tan x = tan alpha`
or `x = tan^(-1)(2//3), pi+tan^(-1)(2//3),tan^(-1)(2//3)-pi, tan^(-1)(2//3)-2pi`