Jeśli dwie proste przecinają się pod kątem prostym ⇒ są prostopadłe ⇒ iloczyn ich współczynników kierunkowych wynosi - 1
[tex]-5x-4y-6=0\\\\-4y=5x+6~~\mid \div (-4)\\\\y=-\frac{5}{4} x-\frac{6}{4} \\\\l:~~y=-\frac{5}{4} x-1\frac{1}{2} ~~gdzie~~a_{1} =-\frac{5}{4} \\\\\\\\y=\frac{m+4}{2} \cdot x -5\\\\y=(\frac{1}{2} m+2)\cdot x -5\\\\k:~~y=(\frac{1}{2} m+2)\cdot x -5 ~~gdzie~~a_{2} =\frac{1}{2} m+2\\\\\\[/tex]
k ⊥ l ⇔ a₁ × a₂ = - 1
[tex]a_{1} \cdot a_{2} = -1 ~~\land ~~a_{1}=-\frac{5}{4} ~~\land~~a_{2}=\frac{1}{2} m+2\\\\-\frac{5}{4} \cdot ( \frac{1}{2} m+2)=-1~~\cdot (-4)\\\\5\cdot ( \frac{1}{2} m+2)=4\\\\\frac{5}{2} m+10=4\\\\\frac{5}{2} m=4-10\\\\\frac{5}{2} m=-6~~\mid \div \frac{5}{2} \\\\m=-6\cdot \frac{2}{5} \\\\m=-\frac{12}{5}\\\\m=-2\frac{2}{5} \\\\m=-2,4[/tex]
Sprawdzenie:
[tex]a_{2} =\frac{m+4}{2} ~~\land ~~m=-2,4~~\Rightarrow~~a_{2} =\frac{-2,4+4}{2}\\\\a_{2} =\frac{-2,4+4}{2}\\\\a_{2} =\frac{-2,4+4}{2}\\\\a_{2} =\frac{-1,6}{2}\\\\a_{2}=0,8\\\\a_{2}=\frac{4}{5} \\\\a_{2}=\frac{4}{5}~~\land~~a_{1}=-\frac{5}{4} ~~\Rightarrow ~~a_{1}\cdot a_{2}=-\frac{5}{4} \cdot \frac{4}{5} =-1~~\Rightarrow~~l \perp k[/tex]
