Find the value of `a` for which the sum of the squares of the roots of the equation `x^2-(a-2)x-a-1=0 ` assumes the least value.

Options

A. 0
B. 1
C. 2
D. 3

Answer

B

Explanation

Let `alpha, beta` be the roots of the given equation. Then, `alpha + beta = a -2 and alpha beta = - (a+1)`.
`therefore" "alpha^(2)+beta^(2)=(alpha+beta)^(2)-2 alpha beta = (a-2)^(2) + 2(a+1)`
`rArr" "alpha^(2) + beta^(2) = a^(2) - 2a + 6 = (a-1)^(2) + 5`
Clearly, `alpha^(2) + beta^(2) ge 5`. So, the minimum value of `alpha^(2)+beta^(2)` is 5 which it attains at = 1.