Question

Statement-1: Number of integral values of $b$ for which the origin and the point $(1,1)$ lie on the same side of the straight line $a^{2}x+aby+1=0∀a∈R$ is three.
Statement-2: If the points $P\left(x_1,y_1\right)$ and $Q\left(x_2,y_2\right)$ lie on the same side of the line $Ax+By+C=0$ then the expressions $A{x}_1+B{y}_1+C$ and $A{x}_2+B{y}_2+C$ will be of the same sign.

• A. Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement-2 is false.
• D. Statement-1 is false, statement-2 is true.

Answer 1

Answer: (A)

$a^2+ab+1>0∀a∈R⇒D<0$
$⇒b^2−4<0⇒b∈(−2,2)$