The equaiton `((x)/(x+1))^(2) + ((x)/(x-1))^(2) = a(a-1)` has

Options

A. four real roots if `a gt 2`
B. four real roots if `a lt -1`
C. two real roots if `1lt a lt 2`
D. no real root if a `lt - 1`

Answer

1,2,3

Explanation

`((x)/(x+1))^(2) +((x)/(x-1))^(2) = a(a-1)`
`or ((x)/(x+1)+(x)/(x-1))^(2) -2((x)/(x+1))((x)/(x-1))) = a (a-1)`
`or ((2x^(2))/(x^(2)-1))^(2) - (2x^(2))/(x^(2)-1)-a(a-1) = 0`
`or t^(2) - t-a(a-1) = 0" ""Where" t= (2x^(2))/(x^(2)-1)`
`rArr t=a or 1 -a`
`rArr (2x^(2))/(x^(2)-1)= aor (2x^(2))/(x^(2)-1)= 1-a`
`rArrx = pm sqrt((a)/(a-2)) or x = pm sqrt((a-1)/(a+1))`
When `alpha lt -1 rArr` all roots are real.
`1 lt a lt 2 rArr x = pm sqrt((a)/(2-a)) i, pm sqrt((a-1)/(a+1))`
`rArr ` two real roots.
When `alpha gt 2 rArr ` allroots are real.