Question

Find $k$ if the lines $2x-3y+4=0$ and $x+ky=3$ are (i) parallel (ii) perpendicular

Answer 1

Slope $m$ of the line $ax+by+c=0$ is,

$m=-\frac{a}{b}$

Slope of the line $2x-3y+4=0$ is $m_1=\frac{2}{3}$

Slope of the line $x+ky=3$ is $m_2=-\frac{1}{k}$

The lines are parallel if

$m_1=m_2$

$\frac{2}{3}=-\frac{1}{k}$

$k=-\frac{3}{2}$

The lines are perpendicular if

$m_1\times m_2=-1$

$\frac{2}{3}(-\frac{1}{k})=-1$

$\Rightarrow k=\frac{2}{3}$