Question

A straight line passes through the point of intersection $x-2y-2=0$ 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 ${45}^{\circ }$

• A. $(-\infty ,4)\cup (7,\infty )$
• B. $(-\infty ,5)\cup (7,\infty )$
• C. $(-\infty ,4)\cup (5,7)\cup (7,\infty )$
• D. $(-\infty ,4)\cup (4,5)\cup (7,\infty )$

Answer 1

The correct option is D $(-\infty ,4)\cup (4,5)\cup (7,\infty )$

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
$-2\lambda -6=0$
$\lambda =-3$
$\therefore -x+(6-b)y=0$
$\frac{1}{6-b}$
As its angle with y = 0 is less than $\frac{\pi }{4}$
$\therefore -1<\frac{1}{6-b}<1$
$\Rightarrow 6-b>1or<-1\Rightarrow b<5orb>7$
But $b\ne 4$ (as the lines WON'T intersect)
$\therefore bϵ(-\infty ,4)\cup (4,5)\cup (7,\infty )$