If `z=sec^(-1)(x+1/x)+sec^(-1)(y+1/y),` where `x y<0,` then the possible values of `z` is (a) `(8pi)/(10)` (b) `(7pi)/(10)` (c) `(9pi)/(10)` (d) `(21pi)/(20)`

Options

A. `(8pi)/(10)`
B. `(7pi)/(10)`
C. `(9pi)/(10)`
D. `(21pi)/(20)`

Answer

C, D

Explanation

`xy lt 0`
`rArr x + (1)/(x) ge 2, y + (1)/(y) le -2`
or `x + (1)/(x) le -2, y + (1)/(y) ge 2`
`x + (1)/(x) ge 2`
`rArr sec^(-1) (x + (1)/(x)) in [(pi)/(3), (pi)/(2))`
`y + (1)/(y) le -2`
`rArr sec^(-1) (y + (1)/(y)) in ((pi)/(2), (2pi)/(3)]`
`rArr z in ((5pi)/(6), (7pi)/(6))`