Question

Line k is perpendicular to line p.the slope of line k is 2/3.if (-1,2) lies on line p,what is the equation of line p?

Answer 1

keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of line P

`\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{2}{3}} ~\hfill \stackrel{reciprocal}{\cfrac{3}{2}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{3}{2} }}`

so we're really looking for the equation of a line whose slope is -3/2 and it passes through (-1 , 2)

`(\stackrel{x_1}{-1}~,~\stackrel{y_1}{2})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{3}{2} \\\\\\ \begin{array} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{- \cfrac{3}{2}}(x-\stackrel{x_1}{(-1)}) \implies y -2 = - \cfrac{3}{2} ( x +1) \\\\\\ y-2=- \cfrac{3}{2}x- \cfrac{3}{2}\implies y=- \cfrac{3}{2}x- \cfrac{3}{2}+2\implies {\Large \begin{array}{llll} y=- \cfrac{3}{2}x+\cfrac{1}{2} \end{array}}`