1. Tardigrade
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  3. Mathematics
  4. 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 a2 x+a b y+1=0 ∀ a ∈ R is three. Statement-2: If the points P(x1, y1) and Q(x2, y2) lie on the same side of the line Ax + By + C =0 then the expressions Ax 1+ By 1+ C and Ax 2+ By 2+ C will be of the same sign.

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Q. 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 $a^2 x+a b y+1=0 \forall a \in R$ 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 $Ax _1+ By _1+ C$ and $Ax _2+ By _2+ C$ will be of the same sign.

Straight Lines

Solution:

$ a^2+a b+1>0 \forall a \in R \Rightarrow D<0$
$\Rightarrow b^2-4<0 \Rightarrow b \in(-2,2)$