The equation of the line perpendicular to the given line that contains point $ A $ is:
$ \color{blue}{ 2x+y=0 }$ ( General form )
$ \color{blue}{ y = - 2 x } ~~~$ ( Slope y-intercept form )
Step 1: The first thing that we need to do is to find the slope of a given line.
The explicit equation of a given line is: $ y = \frac{ 1 }{ 2 } x - 4 $
So the slope is $ m = \frac{ 1 }{ 2 } $.
Step 2: The perpendicular slope ($ m_1 $) is negative reciprocal of the slope $ m $.
$$ m_1 = - \frac{1}{m} = -\frac{ 1 }{ \frac{ 1 }{ 2 } } = -2 $$So the perpendicular line will have a slope of $ m_1 = -2 $
Step 3: Now we have a point and the slope so we can use point-slope form, which is:
$$ y - y_0 = m_1 (x - x_0) $$In this example we have: $ m_1 = -2 $ , $ x_0 = 0 $ and $ y_0 = 0 $. After substitution we have:
$$ \begin{aligned} y - y_0 =& ~ m_1 (x - x_0) \\ y - 0 =& ~ -2 ( x - 0) \\y =& ~ - 2 x\\ \end{aligned} $$