Let S be the set of positive integral values of a for which ax^2+2(a+1)x+9a+4x^2-8x+32

Question

Let $S$ be the set of positive integral values of $a$ for which $\frac{a{x}^{2}+2(a+1)x+9a+4}{{x}^{2}-8x+32}<0,\forallx\in R$. Then, the number of elements in $S$ is :

Accepted Answer

${x}^{2}-8x+32>0\forallx\in R$ as discriminant of this quadratic is $64-4	imes 32<0$

$\Rightarrow a{x}^{2}+2(a+1)x+9a+4<0\forallx\in R$

$\Rightarrow$ Only possible when $a<0$ and $D<0$

$\Rightarrow$ Since $S$ is set of positive
values of $a\Rightarrow S$ is a null set

$\Rightarrow n(S)=0$

Solutions · 1