Question

question 1-11 find the coordinates of point p along the directed line segment ab with a(-4,8) and b(16, -2), so that the ratio of ap to pb is 3 to 2. (8, 2) (6, 3) (4, 4) (2, 5)

Explanation:

Step1: Recall the section formula

The section formula for a point \( P(x,y) \) dividing the line segment joining \( A(x_1,y_1) \) and \( B(x_2,y_2) \) in the ratio \( m:n \) is \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2+ny_1}{m + n} \). Here, \( A(-4,8) \), \( B(16,-2) \), \( m = 3 \), \( n = 2 \).

Step2: Calculate the x - coordinate of P

Substitute \( x_1=-4 \), \( x_2 = 16 \), \( m = 3 \), \( n = 2 \) into the x - coordinate formula:
\( x=\frac{3\times16+2\times(-4)}{3 + 2}=\frac{48-8}{5}=\frac{40}{5}=8 \)

Step3: Calculate the y - coordinate of P

Substitute \( y_1 = 8 \), \( y_2=-2 \), \( m = 3 \), \( n = 2 \) into the y - coordinate formula:
\( y=\frac{3\times(-2)+2\times8}{3 + 2}=\frac{-6 + 16}{5}=\frac{10}{5}=2 \)

Answer:

(8, 2)