Question

If the lines mx - ny + 5 = 0 and 2x + 3y + 6 = 0 are perpendicular to each other, then find the relation between m and n

Answer 1

$\begin{array}{c}mx-ny+5=0\\ \Rightarrow mx+5=ny\\ \Rightarrow ny=mx+5\\ \Rightarrow y=\frac{m}{n}x+\frac{5}{n}\\ \therefore Slopeofline=\frac{m}{n}\\ \Rightarrow m_1=\frac{m}{n}\end{array}|\begin{array}{c}2x+3y+6=0\\ \Rightarrow 3y=-2x-6\\ \Rightarrow y=-\frac{2}{3}x-2\\ \therefore Slopeofline=-\frac{2}{3}\\ \Rightarrow m_2=-\frac{2}{3}\end{array}$
$\because$ Lines are perpendicular
$m_1{m}_2=-1$
$\Rightarrow \frac{m}{n}\times (-\frac{2}{3})=-1$
$\Rightarrow 2m=3n$