Question

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

Answer 1

Hint: The angle $\theta $ between two lines of given slope $m_1$ and $m_2$ is given by the formula-
$tan\theta = \left| {\dfrac{{{{\text{m}}_2} - {{\text{m}}_1}}}{{1 + {{\text{m}}_1}{{\text{m}}_2}}}} \right|$

Complete step-by-step answer:

We have been given the angle between the two lines and the slope of one of the lines. Let the slope of the second line be m. Then applying the formula we can write that-
$\tan \dfrac{{{\pi }}}{4} = \left| {\dfrac{{{\text{m}} - \dfrac{1}{2}}}{{1 + \dfrac{1}{2}{\text{m}}}}} \right|$
$1 = \left| {\dfrac{{2{\text{m}} - 1}}{{2 + {\text{m}}}}} \right|$

We can open the modulus sign and replace it with the plus-minus sign-
$\dfrac{{2{\text{m}} - 1}}{{{\text{m}} + 2}} = \pm 1$
$2{\text{m}} - 1 = {\text{m}} + 2\;\;$
${\text{m}} = 3$
Also,
$2{\text{m}} - 1 = - {\text{m}} - 2$
$3{\text{m}} = - 1$
${\text{m}} = - \dfrac{1}{3}$
Hence, the slope of the other line can be 3 or $\dfrac{-1}{3}$ .

Note: Students often forget to consider both the cases while finding the slope. We should remember that whenever we eliminate the modulus sign, then we need to replace it by the plus-minus sign, hence we get two cases and two answers.