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

1,4

Explanation

`|x_(1) - x_(2)| lt 1`
` therefore (x_(1) + x_(2)) ^(2) - 4 x_(1) x_(2) lt 1`

`rArr (1)/(alpha^(2)) - 4 lt 1 `
`rArr 5 - (1) /(alpha^(2)) gt 0 `
` rArr (5alpha ^(2) - 1)/(alpha ^(2)) gt 0 `
`rArr alpha in (-infty, - (1)/(sqrt(5)) )cup ((1)/(sqrt(5)) , infty)` (1)
Also ` D gt 0 `
` therefore 1 - 4 alpha^(2) gt 0 ` (2)
From (1) and (2)
` alpha in (- (1)/(2) - (1)/(sqrt(5)) )cup ((1)/(sqrt(5)),(1)/(sqrt(5)) )` .