If the roots of the equation `x^2+2a x+b=0` are real and distinct and they differ by at most `2m ,t h e nb` lies in the interval `(a^2,a^2,+m^2)` b. `(a^2-m^2, a 62)` c. `[a^2-m^2, a^2)` d. none of these

Options

A. `(a^(2), a^(2) + m^(2))`
B. `(a^(2) - m^(2), a^(2))`
C. `[a^(2) - m^(2), a^(2))`
D. none of these

Answer

3

Explanation

Let the roots be `alpha, beta`.
`therefore alpha + beta = -2a and alphabeta = b`
Given, `|alpha - beta| le 2m`
or `|alpha - beta|^(2) le (2m)^(2)`
`(alpha + beta)^(2) - 4ab le 4m^(2)`
or `4a^(2) - 4b le 4m^(2)`
`rArr a^(2) - m^(2) le b` and discriminant `D gt 0` or `4a^(2) - 4b gt 0`
`rArr a^(2) - m^(2) le b and b le a^(2)`
Hence, `b in [a^(2) - m^(2), a^(2)).`