Question

The 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

• A. $1$
• B. $3$
• C. $2$
• D. $5$

Answer 1

The correct option is B $3$

For same side of $a^{2}x+aby+1=0$ of origin & $(1,1).$
$1\times \frac{a^2+ab+1}{1}>0$
Given, $a\in R-\left\{0\right\}$
Then,
$a^2+ab+1>0$
This must be true for all values of $a$ except 0.
Hence,
$\Delta >0$ and coefficient of $a^2$ must be positive.
$\Rightarrow b^2-4<0$
$\Rightarrow |b|<2$
The integral values which $b$ can take are $-1,0,1$