If the angle between the lines is $\frac{\pi^c}{4}$ and slope of one of the lines is $\frac{1}{2}$, then…

If the angle between the lines is $\frac{\pi^c}{4}$ and slope of one of the lines is $\frac{1}{2}$, then slope of the other line is
  1. 3 or $-\frac{1}{3}$
  2. 4 or $-\frac{1}{4}$
  3. 2 or $-\frac{1}{2}$
  4. 3 or -3

Solution

We have $\theta=45^{\circ}$ and $\mathrm{m}_1=\frac{1}{2}$ $\begin{aligned} & \tan \theta=\left|\frac{\mathrm{m}_1-\mathrm{m}_2}{1+\mathrm{m}_1 \mathrm{~m}_2}\right| \\ & \therefore \quad \tan 45^{\circ}=\left|\frac{\frac{1}{2}-\mathrm{m}_2}{1+\left(\frac{1}{2}\right) \mathrm{m}_2}\right| \Rightarrow\left|\frac{1-2 \mathrm{~m}_2}{2+\mathrm{m}_2}\right|= \pm 1 \\ & \therefore \quad 1-2 \mathrm{~m}_2=2+\mathrm{m}_2 \text { or } 1-2 \mathrm{~m}_2=-2 \\ & \therefore \mathrm{m}_2=\frac{-1}{3} \text { or } \mathrm{m}_2=3 \end{aligned}$

Asked in: MHT CET 2021 (23 Sep Shift 1)

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