Question

Find the co-ordinates of those points on the line 3x+2y=5 which are equidistant from the lines 4x+3y=7 and 2y-5=0

Answer 1

$$4x + 3y = 7 \rightarrow l_1 = \frac{4x + 3y - 7}{\sqrt{16 + 9}}$$ $$2y - 5 = 0 \rightarrow l_2 = \frac{2y - 5}{\sqrt{4}}$$ $$3x + 2y = 5 \rightarrow y = \frac{5}{2} - \frac{3}{2}x$$

Substituting $y = \frac{5}{2} - \frac{3}{2}x$ in $l_1$ and $l_2$, and $|l_1| = |l_2|$ or $|l_1| = -|l_2|$

Answer: $\left(-\frac{1}{14}, \frac{73}{28}\right), \left(\frac{1}{16}, \frac{77}{32}\right)$