Question

If expression $x^2-4cx+b^2>0$for all real values of $x$ and $a^2+b^2<ab$, then range of the function $\frac{x+a}{x^2+bx+c^2}$ is

• A. $(-\infty ,0)$
• B. $(0,\infty )$
• C. $(-\infty ,\infty )$
• D. None of these

Answer 1

The correct option is C $(-\infty ,\infty )$

$\begin{array}{c}Letis,\\ Wehavetheequ:x^2-4cx+b^2>0,\\ anditsdiscriminateis:D<0\\ \Rightarrow 16c^2-4b^2<0\\ \Rightarrow 4c^2-b^2<0\\ Now,\\ \frac{x+a}{x^2+bx+c^2},anddiscriminateof{x}^2+bx+c^2\\ \Rightarrow b^2-4c^2>0,soitsrootsa\&\beta \\ thenexpressionof\frac{x+a}{(x-\alpha )(x-\beta )}[wecanunderstandthroughgraph\\ Inthisconditionthewholelineattendsto-\infty to+\infty .\\ so,wecansaytherangeofy=\left(-\infty ,\infty \right)arerealno.andthecorrectoptionisC.\end{array}$