Question

1. given (overline{ab}) with (a(8, -4)) and (b(-6, -11)), if (p) partitions (ab) such that the ratio of (ap) to (ab) is (2:7), find the coordinates of (p).

Explanation:

Step1: Understand 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, \( AP:AB = 2:7 \), so \( AP:PB=2:(7 - 2)=2:5 \). Thus, \( m = 2 \), \( n = 5 \), \( x_1 = 8 \), \( y_1=-4 \), \( x_2=-6 \), \( y_2 = - 11 \).

Step2: Calculate the x - coordinate of P

Substitute the values into the formula for \( x \):
\[
x=\frac{2\times(-6)+5\times8}{2 + 5}=\frac{-12 + 40}{7}=\frac{28}{7}=4
\]

Step3: Calculate the y - coordinate of P

Substitute the values into the formula for \( y \):
\[
y=\frac{2\times(-11)+5\times(-4)}{2+5}=\frac{-22-20}{7}=\frac{-42}{7}=-6
\]

Answer:

The coordinates of \( P \) are \( (4,-6) \)