Question

If the angle between two lines is $\frac{\pi }{4}$ and slope of one of the lines is $\frac{1}{2}$, find the slope of the other line.

• A. $3$ or $\frac{-1}{3}$
• B. $2$ or $\frac{-1}{2}$
• C. $4$ or $\frac{-1}{4}$
• D. $3$ or $-3$

Answer 1

Answer: (A)

We know that the acute angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by
$tan\theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|\dots \left(i\right)$
Let $m_1=\frac{1}{2}$,
$m_2=m$ and
$\theta =\frac{\pi }{4}$.
Now, putting these values in $\left(i\right)$, we get
$tan\frac{\pi }{4}=\left|\frac{m-\frac{1}{2}}{1+\frac{1}{2}m}\right|$
$\Rightarrow 1=\left|\frac{2m-1}{2+m}\right|$,
wihich gives $\frac{2m-1}{2+m}=1$ or $\frac{2m-1}{2+m}=-1$
Therefore $m=3$ or $m=-\frac{1}{3}$
Hence, slope of the other line is $3$ or $-\frac{1}{3}$.