Question

A straight line passes through the point of intersection of the lines x - 2y - 2 = 0 and 2x - by - 6 = 0 and the origin then the complete set of values of b for which the acute angle between this line and y = 0 is less than ${45}^0$

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

.As line passes through the point of intersection of x - 2y - 2 = 0 and 2x - by - 6 = 0
It can be represented as $\lambda (x-2y-2)+(2x-by-6)=0$
As it passes through the origin
$-2\lambda -6=0$
$\lambda =-3$
$\therefore$ equation of the line is -x + (6-b) y = 0 Its slope is $\frac{1}{6-b}$
As its angle with y = 0 is less than $\frac{\pi }{4}$
$\therefore -1<\frac{1}{6-b}<1$
$\Rightarrow 6-b>1$ or $<-1\Rightarrow b<5$ or $b>7$
But $(b\ne 4)$ (as the lines intersect)
$\therefore b\in (-\infty ,4)\cup (4,5)\cup (7,\infty )$