The distance of a point $(2,5)$ from the line $3 x+y+4=0$ measured along the line $\mathrm{L}_1$ and…

The distance of a point $(2,5)$ from the line $3 x+y+4=0$ measured along the line $\mathrm{L}_1$ and $\mathrm{L}_2$ are same. If slope of line $L_1$ is $\frac{3}{4}$, then slope of the line $\mathrm{L}_2$ is
  1. $\frac {-3}{4}$
  2. $\frac {1}{3}$
  3. $\frac {1}{4}$
  4. $0$

Solution

According to the given condition, in $\triangle \mathrm{ABC}$, $\mathrm{AB}=\mathrm{AC}$. $\therefore \quad \triangle \mathrm{ABC}$ is an isosceles triangle. Let $\mathrm{m}, \mathrm{m}_1, \mathrm{~m}_2$ be the slopes of given line, $\mathrm{L}_1$ and $\mathrm{L}_2$ respectively. $\begin{array}{ll} \therefore & \mathrm{m}=-3, \mathrm{~m}_1=\frac{3}{4} \\ \therefore & \left|\frac{\mathrm{m}-\mathrm{m}_1}{1+\mathrm{mm}_1}\right|=\left|\frac{\mathrm{m}-\mathrm{m}_2}{1+\mathrm{mm}_2}\right| \\ \therefore & 3=\left|\frac{-3-\mathrm{m}_2}{1+3 \mathrm{~m}_2}\right| \\ & \Rightarrow \mathrm{m}_2=0 \end{array}$

Asked in: MHT CET 2023 (09 May Shift 1)

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