Question

determine the following. *

1. find the coordinates of point p along the directed line segment ab with a(-3,2) and b(5, -4), so that p partitions ab in the ratio 2 to 6.

Explanation:

Step1: Recall the section formula

The section formula for a point \( P(x,y) \) that divides the line segment joining \( A(x_1,y_1) \) and \( B(x_2,y_2) \) in the ratio \( m:n \) is given by:

$$ x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n} $$

Here, \( A(-3,2) \), \( B(5,-4) \), \( m = 2 \), \( n = 6 \).

Step2: Calculate the x - coordinate of P

Substitute \( x_1=-3 \), \( x_2 = 5 \), \( m = 2 \), \( n = 6 \) into the formula for \( x \):

$$ x=\frac{2\times5+6\times(-3)}{2 + 6}=\frac{10-18}{8}=\frac{-8}{8}=-1 $$

Step3: Calculate the y - coordinate of P

Substitute \( y_1 = 2 \), \( y_2=-4 \), \( m = 2 \), \( n = 6 \) into the formula for \( y \):

$$ y=\frac{2\times(-4)+6\times2}{2 + 6}=\frac{-8 + 12}{8}=\frac{4}{8}=\frac{1}{2} $$

Answer:

The coordinates of point \( P \) are \( (-1,\frac{1}{2}) \)