Question

Find the equation of a line that has $y$-intercept as $4$ and is perpendicular to line joining $(2,-3)$ and $(4,2)$.

Answer 1

Slope of line joining $(2,-3)$ and $(4,2)=\frac{(2+3)}{(4-2)}=\frac{5}{2}$

Equation of a line with slope $m$ and $y$ -intercept $‘c^{\prime }$ is $y=mx+c$.

$c=4$, slope ($m$) $=\frac{-1}{\frac{5}{2}}=-\frac{2}{5}$

[$\because$ the lines are perpendicular, product of slopes two perpendicular lines $=-1$]

So, $y=(-\frac{2}{5})x+4$

$\Rightarrow y=\frac{-2x+20}{5}$

$\Rightarrow 5y=-2x+20$

$\Rightarrow 2x+5y-20=0$

Hence, the equation of a line that has $y$-intercept as $4$ and is perpendicular to line joining $(2,-3)$ and $(4,2)$ is $2x+5y-20=0.$