Let `alpha and beta` be the roots of the equation
`x^(2) - ( 1 - 2a^(2)) x + ( 1 - 2a^(2) )= 0 `
Under what condition is `1/alpha ^(2) + 1/beta^(2) lt1` ?

Options

A. `a^(2) lt 1/2`
B. `a^(2) gt 1/2`
C. `a^(2) gt 1 `
D. `a^(2) in (1/3,1/2) ` omly

Answer

A

Explanation

` 1/alpha^(2) + 1/beta^(2) lt 1 rArr (alpha^(2) + beta^(2))/(alpha beta)^(2) lt1`
` rArr ((alpha + beta)^(2) - 2 alpha beta)/(alpha beta)^(2) lt 1 ` ` alpha + beta = (-b)/ a = ( 1 - 2a^(2)) & alpha beta = c/a = ( 1 - 2a^(2))`
On solving : ` (4a^(4)- 1)/(4a^(4) - 4a^(2) + 1 ) lt1`
` (4a^(4)- 1)/(4a^(4) - 4a^(2) + 1 ) lt1`
`4a^(2) lt 2 rArr a^(2) lt 1/2 `