Question

If the angle between two lines is $\frac{\pi }{4}$ and slope of the lines is $\frac{1}{2}$, then which of the following is not the slope of the other line?

• A. 3
• B. $\frac{-1}{3}$
• C. $\frac{1}{3}$
• D. None of these

Answer 1

Answer: (C)

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

Hence, slope of the other line is 3 or $-\frac{1}{3}$.