If `a, b, c` are distinct positive real numbers such that the quadratic expression `Q_(1)(x) = ax^(2) + bx + c`, `Q_(2)(x) = bx^(2) + cx + a, Q_(3)(x) = cx^(2) + x + b` are always non-negative, then possible integer in the range of the expression `y = (a^(2)+ b^(2) + c^(2))/(ab + bc + ca)` is

Options

A. `1`
B. `2`
C. `3`
D. `4`

Answer

B, C

Explanation

For `a ne b ne c a, b, c in R^(+)`
if `ax^(2) + bx + c ge 0` then `a gt 0` and `b^(2) - 4ac le 0`
`bx^(2) + cx + a ge 0` then `b gt 0` and `c^(2) - 4ab le 0`
`cx^(2) + ax + b ge 0` then `c gt 0` and `a^(2) - 4bc le 0`
`rArr a^(2)+b^(2)+c^(2)-4(ab+bc+ac)le0 rArr (a^(2)+b^(2)+c^(2))/(ab+bc+ca)lt4`
Also we know that `(a-b)^(2) + (b-c)^(2) + (c-a)^(2) gt 0`
`a^(2) + b^(2) + c^(2) - ab - bc- ac gt 0 rArr (a^(2) + b^(2) + c^(2))/(ab + bc + ca) gt 1`
`:.` range is `(1, 4)`