Question

P along the directed line segment AB AB , from A(1, 6) A(1, 6) to B(−2,−3) B(−2,−3) , so that the ratio of AP AP to PB PB is 5 5 to 1 1 .

Answer 1

assuming we're looking for point P.

`\textit{internal division of a line segment using ratios} \\\\\\ A(1,6)\qquad B(-2,-3)\qquad \qquad \stackrel{\textit{ratio from A to B}}{5:1} \\\\\\ \cfrac{A\underline{P}}{\underline{P} B} = \cfrac{5}{1}\implies \cfrac{A}{B} = \cfrac{5}{1}\implies 1A=5B\implies 1(1,6)=5(-2,-3)`

`(\stackrel{x}{1}~~,~~ \stackrel{y}{6})=(\stackrel{x}{-10}~~,~~ \stackrel{y}{-15}) \implies P=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{1 -10}}{5+1}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{6 -15}}{5+1} \right)} \\\\\\ P=\left( \cfrac{ -9 }{ 6 }~~,~~\cfrac{ -9}{ 6 } \right)\implies P=\left(-\cfrac{3}{2}~~,~-\cfrac{3}{2} \right)\implies P=\left(-1(1)/(2)~~,~ -1(1)/(2) \right)`