The equation of the line perpendicular to the given line that contains point $ A $ is:
$ \color{blue}{ 9x-10y+12=0 }$ ( General form )
$ \color{blue}{ y = \frac{ 9 }{ 10 } x + \frac{ 6 }{ 5 } } ~~~$ ( Slope y-intercept form )
Step 1:The slope of a given line is $ m = -\frac{ 10 }{ 9 } $.
Step 2: The perpendicular slope ($ m_1 $) is negative reciprocal of the slope $ m $.
$$ m_1 = - \frac{1}{m} = -\frac{ 1 }{ -\frac{ 10 }{ 9 } } = \frac{ 9 }{ 10 } $$So the perpendicular line will have a slope of $ m_1 = \frac{ 9 }{ 10 } $
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 = \frac{ 9 }{ 10 } $ , $ x_0 = 2 $ and $ y_0 = 3 $. After substitution we have:
$$ \begin{aligned} y - y_0 =& ~ m_1 (x - x_0) \\ y - 3 =& ~ \frac{ 9 }{ 10 } ( x - 2) \\y -3 =& ~ \frac{ 9 }{ 10 } x -\frac{ 9 }{ 5 } \\y =& ~ \frac{ 9 }{ 10 } x -\frac{ 9 }{ 5 } + 3 \\y =& ~ \frac{ 9 }{ 10 } x + \frac{ 6 }{ 5 }\\ \end{aligned} $$