If $x^2-4 c x+b^20 \forall x \in R$ and $a^2+c^2

Question

If $x^2-4 c x+b^2>0 \forall x \in R$ and $a^2+c^2<a b$ then range of $f(x)=\frac{x+a}{x^2+b x+c^2}$ is

Professor PC Thomas Classes · Answer

$x^{2}-4cx+b^{2}>0$ coefficient $x^{2}=1>0 \Rightarrow$ disc. $<0 \Rightarrow 16c^{2}-4b^{2}<0 \Rightarrow 4c^{2}0$ and Disc. $=(4a-2b)^{2}-4(b^{2}-4c^{2}) = 16(a^{2}+c^{2}-ab)<0$ from (2). $ herefore y \in R$. Range of $f(x)$ is $(-\infty, +\infty)$.