Question

Number of integral values of $b$ for which the origin and the point $(1,1)$ lie on the same side of the straight line $a^{2}x+aby+1=0$ for all $a\in R-\left\{0\right\}$ is

Answer 1

$Equationofline\to a^{2}x+aby+1=0\left[given\right]$

$\begin{array}{c}forpt\left(0,0\right)\to 0+0+1\Rightarrow 1>0\\ forpt\left(1,1\right)\to a^2\left(1\right)+ab+1>0\end{array}$

No Real roots of quadratic equation.

So,

$\begin{array}{c}D<0\\ b^2-4<0\\ b^2<4whenb\in \left(-2,2\right)\end{array}$

So, value of positive value $[0,2]$