Question

1. given (overline{df}) with (d(-1, 11)) and (f(-9, -5)), if (e) partitions (df) such that the ratio of (de) to (df) is (5:8), find the coordinates of (e).

Explanation:

Step1: Identify the section formula

The section formula for a point \( E(x,y) \) that divides the line segment joining \( D(x_1,y_1) \) and \( F(x_2,y_2) \) in the ratio \( m:n \) is \( x=\frac{mx_2 + nx_1}{m + n} \), \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( DE:DF = 5:8 \), so \( DE:EF=5:(8 - 5)=5:3 \), \( m = 5 \), \( n = 3 \), \( x_1=-1,y_1 = 11,x_2=-9,y_2=-5 \).

Step2: Calculate the x - coordinate of E

Substitute into the x - formula: \( x=\frac{5\times(-9)+3\times(-1)}{5 + 3}=\frac{-45-3}{8}=\frac{-48}{8}=-6 \)

Step3: Calculate the y - coordinate of E

Substitute into the y - formula: \( y=\frac{5\times(-5)+3\times11}{5 + 3}=\frac{-25 + 33}{8}=\frac{8}{8}=1 \)

Answer:

The coordinates of \( E \) are \((-6,1)\)