Question

find the coordinates of point p along the directed line segment ab, from a(-3,2) to b(5,-4), so that the ratio of ap to pb is 2 to 6. the coordinates are p(□,□).

Explanation:

Step1: Calculate the x - coordinate of P

Use the section - formula for x - coordinate $x=\frac{m_1x_2 + m_2x_1}{m_1 + m_2}$, where $m_1 = 2$, $m_2=6$, $x_1=-3$, $x_2 = 5$.
$x=\frac{2\times5+6\times(-3)}{2 + 6}=\frac{10-18}{8}=\frac{-8}{8}=-1$

Step2: Calculate the y - coordinate of P

Use the section - formula for y - coordinate $y=\frac{m_1y_2 + m_2y_1}{m_1 + m_2}$, where $m_1 = 2$, $m_2=6$, $y_1 = 2$, $y_2=-4$.
$y=\frac{2\times(-4)+6\times2}{2 + 6}=\frac{-8 + 12}{8}=\frac{4}{8}=\frac{1}{2}$

Answer:

$P(-1,\frac{1}{2})$