Let `S` be the set of all non-zero real numbers such that the quadratic equation `alphax^2-x+alpha=0` has two distinct real roots `x_1a n dx_2` satisfying the inequality `|x_1-x_2|<1.` Which of the following intervals is (are) a subset (s) of `S ?` `(1/2,1/(sqrt(5)))` b. `(1/(sqrt(5)),0)` c. `(0,1/(sqrt(5)))` d. `(1/(sqrt(5)),1/2)`

Options

A. `(-(1)/(2),-(1)/(sqrt(5)))`
B. `(-(1)/(sqrt(5)),0)`
C. `(0,(1)/(sqrt(5)))`
D. `((1)/(sqrt(5)),(1)/(2))`

Answer

D

Explanation

It is given that `x_(1), x_(2)` are distinct real roots of `ax^(2) - x + alpha = 0`.
`therefore" "x_(1)+x_(2)=(1)/(alpha) and x_(1) x_(2) = 1`.
Now, `|x_(1)+x_(2)|lt 1`
`rArr" "0 lt (x_(1)-x_(2))^(2) lt 1`
`rArr" "0 lt (x_(1)+x_(2))^(2) - 4 x_(1) x_(2) lt 1`
`rArr" "0 lt (1)/(alpha^(2))-4 lt 1`
`rArr" "0 lt(1-4 alpha^(2))/(alpha^(2)) lt 1`
`rArr" "(1-4 alpha^(2))/(alpha^(2)) gt 0 and (1-4 alpha^(2))/(alpha^(2)) lt 1`
`rArr" "(4 alpha^(2)-1)/(alpha^(2)) lt 0 and 1 - 5 alpha^(2) lt 0`
`rArr" "4 alpha^(2) -1 lt 0 and 5 alpha^(2) -1 gt o`
`rArr" "alpha in (-1//2, 1//2) and alpha in (-oo, -1// sqrt(5)) uu (1// sqrt(5), oo)`
`rArr" "alpha in (-1//2, -1// sqrt(5)) uu (1// sqrt(5), 1//2)`