Question

If $\forall \lambda \in R$, equation $a{x}^2+(b-\lambda )x+(a-b-\lambda )=0,a\ne 0,a,b\in R$ has real roots, then which of the following may be true?

• A. $a=b$
• B. $0>a>b$
• C. $b\le a<0$
• D. $a>b>0$

Answer 1

Answer: (C)

$b\le a<0$

$a{x}^2+(b-\lambda )x+(a-b-\lambda )=0$
roots are real
$D\ge 0$
$\Rightarrow {\lambda }^2+2(2a-b)\lambda +b^2-4a^2+4ab\ge 0\forall \lambda \in R$
$\Rightarrow D_1\le 0$
$a(a-b)\le 0$
$b\le a<0$