Question
Let $L_{1}:3x + 4y = 1$ and $L_{2}:5x - 12y + 2 = 0$ be two given lines. Let image of every point on $L_{1}$ with respect to a line $L$ lies on $L_{2}$ then possible equation of $L$ can be:
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Solution
Sol. [Key Concept: The mirror $L$ must be an angle bisector of $L_{1}$ and $L_{2}$; there are two bisectors.]
$L$ must be angle bisector of $L_{1}$ and $L_{2}$
$\therefore\ L$ is given by
$$\begin{matrix} \frac{3x + 4y - 1}{5} = \pm \frac{5x - 12y + 2}{13} \\ \Rightarrow \ 14x + 112y - 23 = 0, \\ 64x - 8y - 3 = 0 \end{matrix}$$
✔ (a) $4x + 112y - 23 = 0$ is one bisector of $L_{1,}L_{2}$.
✔ (b) $64x - 8y - 3 = 0$ is the other bisector.
✘ (c) $11x - 4y = 0$ is not a bisector of the pair.
✘ (d) $52y - 45x = 7$ is not a bisector of the pair.
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