If `alpha and beta` are roots of the equation `x^(2)+5|x|-6=0,` then the value of `|tan^(-1)alpha-tan^(-1)beta|` is

Options

A. `(pi)/(2)`
B. `0`
C. `pi`
D. `(pi)/(4)`

Answer

A

Explanation

Given `alpha and beta` be the roots of the equation
`x^(2)+5|x|-6=0`
Now, `|x|^(2)+5|x|-6=0`
Now, `|x|^(2)+5|x|-6=0`
`|x|^(2)+6|x|-|x|-6=0` . . [by factorisation]
`|x|(|x|+6)-1(|x|6)=0`
`(|x|+6)(|x|-1)=0`
`|x|=-6` or `|x|=1`
(Since, modulus cannot be giving negative values)
`therefore|x|=1implies x=+-1`
So, `alpha=1 and beta=-1`
`therefore`Now, `|tan^(-1)alpha-tan^(-1)beta|=|tan^(-1)1-tan^(-1)(-1)|`
`=|(pi)/(4)-(-(pi)/(4))|=|(pi)/(4)+(pi)/(4)|=|(pi)/(2)|`