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Q.
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.
Straight Lines
Solution:
We know that the acute angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by
$tan\,\theta=\left|\frac{m_{2}-m_{1}}{1+m_{1}m_{2}}\right|\quad\ldots\left(i\right)$
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 $-\frac{1}{3}$.