Question

1. given \\(\overline{ac}\\) with \\(a(3, 4)\\) and \\(c(-9, -2)\\), if \\(b\\) partitions \\(ac\\) such that the ratio of \\(ab\\) to \\(bc\\) is \\(1:5\\), find the coordinates of \\(b\\).

Explanation:

Step1: Recall the section formula

The coordinates of a point \( B(x,y) \) that divides the line segment joining \( A(x_1,y_1) \) and \( C(x_2,y_2) \) in the ratio \( m:n \) are given by \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2+ny_1}{m + n} \). Here, \( A(3,4) \), \( C(-9,-2) \), \( m = 1 \), \( n=5 \).

Step2: Calculate the x - coordinate of B

Substitute \( x_1 = 3 \), \( x_2=-9 \), \( m = 1 \), \( n = 5 \) into the formula for \( x \):
\( x=\frac{1\times(-9)+5\times3}{1 + 5}=\frac{-9 + 15}{6}=\frac{6}{6}=1 \)

Step3: Calculate the y - coordinate of B

Substitute \( y_1 = 4 \), \( y_2=-2 \), \( m = 1 \), \( n = 5 \) into the formula for \( y \):
\( y=\frac{1\times(-2)+5\times4}{1+5}=\frac{-2 + 20}{6}=\frac{18}{6}=3 \)

Answer:

The coordinates of \( B \) are \( (1,3) \)