If `a >0` and discriminant of `a x^2+2b x+c` is negative, then `|[a,b,ax+b],[b,c,bx+c],[ax+b,bx+c,0]|` is a.`+v e` b. `(a c-b)^2(a x^2+2b x+c)` c. `-v e` d. `0`

Options

A. positive
B. `(ac -b^(2)) (ax^(2) + 2bx +c)`
C. negative
D. 0

Answer

C

Explanation

It is given that the discriminant of `ax^(2) + 2bx + c` is negative.
`:. 4b^(2) - 4ac lt 0 rArr b^(2) - ac lt 0 rArr ac -b^(2) gt 0`
Also, `a gt 0` and discriminant is negative.
`:. Ax^(2) + 2bx + c gt 0` for all `x in R`
Now,
`Delta = |(a,b,ax + b),(b,c,bx + c),(ax +b,bx+c,0)|`
`rArr Delta = |(a,b,0),(b,c,0),(ax +b,bx +c,- x (ax +b) - (bx +c))|`
`rArr Delta = |(a,b,0),(b,c,0),(ax +b,bx +c,-(ax^(2) + 2bx +c))|`
`rArr Delta = -(ax^(2) + 2bx + c) (ac -b^(2)) lt 0` [Using (i) and (ii)]