Let `a ,b ,c` be real. If `a x^2+b x+c=0` has two real roots `alphaa n dbeta,w h e r ealpha<<-1a n dbeta>>1` , then show that `1+c/a+|b/a|<0`

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

A

Explanation

Let `f(x) = ax^(2) + bx + c`. Clearly, statement-2 is true (See Theory). Since 1 and -1 lie between the roots and sign of f(x) is opposite to the sign of 'a' for all x lying between the roots (using statement-2).
`therefore" "a f(1) lt 0 and a f(-1) lt 0`
`rArr" "a(a+b+c) lt 0 and a(a-b+c) lt 0`
`rArr" "a^(2)(1+(b)/(a)+(c)/(a))lt 0 and a^(2)(1-(b)/(a)+(c)/(a))lt 0`
`rArr" "1+(b)/(a)+(c)/(a) lt 0 and 1 - (b)/(a)+(c)/(a)lt 0 rArr 1 + |(b)/(a)|+(c)/(a) lt 0`