If the sum of the squares of the roots of the equation `x^(2) - (a-2) x - (a + 1) = 0` is least, then the value of a, is

Options

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

Answer

B

Explanation

Let `alpha` and `beta` be the roots of the equation
`x^(2)-(a-2)x-a+1=0`
Then `alpha+beta=a-2, alpha beta =-a+1`
Let `z= alpha^(2)+beta^(2)=(alpha+beta)^(2)-2alpha beta`
`=(a-2)^(2)+2(a-1)=a^(2)-2a+2`
`implies(dz)/(da)=2a-2`
Put `(dz)/(da)=0`
`impliesa=1`
`:.(d^(2)z)/(da^(2))=2gt0`
So, `z` has minima at `a=1`
So `alpha^(2)+beta^(2)` has least value for `a=1`.