If quadratic equation `ax^(2) + bx + ab + bc + ca - a^(2) - b^(2) - c^(2) = 0` where `a, b, c` distinct reals, has imaginary roots than (A) `a+b+ab+bc+calta^(2)+b^(2)+c^(2)` (B)`a-b+ab+bc+cagta^(2)+b^(2)+c^(2)` (C)`4a+2b+ab+bc+calta^(2)+b^(2)+c^(2)` (D)`2(a+-3b)-9{(a-b)^(2)+(b-c)^(2)+(c-a)^(2)}lt0`

Options

A. `a+b+ab+bc+calta^(2)+b^(2)+c^(2)`
B. `a-b+ab+bc+cagta^(2)+b^(2)+c^(2)`
C. `4a+2b+ab+bc+calta^(2)+b^(2)+c^(2)`
D. `2(a+-3b)-9{(a-b)^(2)+(b-c)^(2)+(c-a)^(2)}lt0`

Answer

A, C, D

Explanation

`f(x) = ax^(2) + bx + ab + bc + ca - a^(2) - b^(2) - c^(2)`
`= ax^(2) + bx - (1)/(2) {(a - b)^(2) + (b + c)^(2) + (c - a)^(2)}`
`:' f(x)` has imaginary roots and
`f(0) = -1/2{(a-b)^(2) - (b-c)^(2)+(c-a)^(2)}`
`f(x) lt 0 AA x in R`
`f(1) lt 0` options `(A)` is correct
`f(1) lt 0` options `(B)` is incorrect `f(Z) lt 0` options `(C)` is correct
`f(+-(1)/(3)) lt 0` options `(D)` is correct