First slide

Equation of line in Straight lines

Question

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

Moderate

A

0
B

1
C

2
D

3

Solution

$(0, 0) and (1, 1) lie on same side of line a^{2} x + a b y + 1 = 0$
$\begin{array}{l} For (0, 0), L : a^{2} (0) + a b (0) + 1 > 0 \\ ∴ for (1, 1), we must have \\ a^{2} + a b + 1 > 0 \forall a \in R \\ ∴ D < 0 \\ ∴ b^{2} - 4 < 0 \\ \Rightarrow b \in (- 2, 2) \\ but b > 0 \Rightarrow b \in (0, 2) \\ T h e n u m b e r o f i n t e g r a l v a l u e s t a k e n b y b i s 1 i e; b = 1 \end{array}$

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