Question

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

• A. $1$
• B. $3$
• C. $2$
• D. $5$

Answer 1

The correct option is C $2$

Putting $(0,0)$ in $a^{2}x+aby+1,$ we get $1>0$
$(1,1)$ and origin should lie on same side of $a^{2}x+aby+1=0$
$\Rightarrow 1\times a^2+ab+1>0$
$\Rightarrow a^2+ab+1>0$

For all values of $a$ except $0,$ $a^2+ab+1>0$
$\Rightarrow D<0$
$\Rightarrow b^2-4<0$
$\Rightarrow -2<b<2$
So, possible non-negative integral values of $b$ are $0,1$