Question

What is the equation of the line perpendicular to y=−32x that passes through (2,−4)?

Answer 1

$\textsf{\Large{\underline{Straight Lines}}}$ :

$\textsf{\underline{Correction}}$ : $\mathsf{y \:=\:{\dfrac{-3}{2}x}}$

$\textsf{\underline{Solution}}$ :

Given, y = $\mathsf{\dfrac{-3}{2}x}$

Points =$\mathsf{( \:2,\: - 4\:)}$

$\mathsf{x\: = \:2}$ and $\mathsf{ y\: =\: - \:4}$

Comparing the given equation with slope intercept - form,

$\boxed{\mathsf{y\:=\:mx\:+\:c}}$

We get,

$\mathsf{m \:= \:{\dfrac{-3} {2}}}$

Let $\mathsf{m_1 \:= \:{\dfrac{-3} {2}}}$

For this we have to find the Slope of given equation,

For Slope of Perpendicular line,

$\boxed{\mathsf{m_1\:m_2\:=\:-1}}$

Here, $\mathsf{m_1\:=\:{\dfrac {-3} {2}}}$

$\mathsf{m_2\:=\:m}$

Now putting these values in above formula of finding slope,

$\mathsf{\dfrac{-3}{2}\:m\:=\:-1}$

➡️ $\mathsf{m\:=\:{\dfrac{2}{3}}}$

Equation of perpendicular line,

$\mathsf{y\:=\:mx\:+\:c}$

$\mathsf{-4\:=\:{\dfrac{2*2}{3}\:+\:c}}$

$\mathsf{c\:=\:-4\:-\:{\dfrac{4}{3}}}$

$\mathsf{c\:=\:-4\:-\:{\dfrac{4}{3}}}$

$\mathsf{c\:=\:{\dfrac{-16}{3}}}$

Putting this value of c and m in general equation of slope - intercept form,

$\mathsf{y\:=\:mx\:+\:c}$

$\mathsf{y\:=\:{\dfrac{2}{3}x\:-\:{\dfrac{16}{3}}}}$

$\mathsf{3y\:=\:2x\:-\:16}$

$\boxed{\mathsf{2x\:-\:3y=\:16}}$