Question

If the slope of the first line is half the slope of the second line and the tangent of the angle between them is $\frac{1}{4}$, then the slope of the first line can be

• A. $\frac{-2-\sqrt{2}}{2}$
• B. $\frac{-2+\sqrt{2}}{2}$
• C. $\frac{2-\sqrt{2}}{2}$
• D. $\frac{2+\sqrt{2}}{2}$

Answer 1

Answer: (A), (B), (C), (D)

$\frac{2+\sqrt{2}}{2}$

Let $m$ be the slope of the first line.
Then slope of another line $=2m$
and $\theta$ is angle between them.
So, $\tan \theta =\left|\frac{2m-m}{1+2m^2}\right|$
$\frac{1}{4}=\pm \frac{m}{1+2m^2}$
$\Rightarrow 1+2m^2=\pm 4m$
$\Rightarrow 2m^2\pm 4m+1=0$
$\Rightarrow m=\frac{4\pm \sqrt{8}}{4},\frac{-4\pm \sqrt{8}}{4}$
$\Rightarrow m=\frac{2\pm \sqrt{2}}{2},\frac{-2\pm \sqrt{2}}{2}$