Question

Find the equation of a line passing through $(2,3)$ and perpendicular to the lines $3x+2y-1=0.$

Answer 1

Line passes through $(2,3)$

Let the be $y=mx+c$ - $\left(A\right)$

$3=2m+c$ - $\left(1\right)$

perpendicular to $3x+2y-1=0$

$\Rightarrow y=\frac{-3x}{2}+\frac{1}{2}$ - $\left(B\right)$

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

$A$ & $B$ are perpendicular

$\Rightarrow m_1\times m_2=-1$

$\Rightarrow m\times \frac{-3}{2}=-1$

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

put $m=2/3$ in $1$

$\Rightarrow 3=2\times \frac{2}{3}+c$

$\Rightarrow 3=\frac{4}{3}+c\Rightarrow c=3-\frac{4}{3}=\frac{5}{3}$

$\therefore y=\frac{2}{3}x+\frac{5}{3}$

$\Rightarrow 3y=2x+5\Rightarrow 2x-3y+5=0$

equation of line passing through $(2,3)$ and perpendicular to $3x+2y-1=0$ is $2x-3y+5=0$