Question

point a is at (-3, 4) and point c is at (2, -6). find the coordinates of point b on \\(\overline{ac}\\) such that the ratio of ab to ac is 4 : 5. b = (\\(\square\\), \\(\square\\))

Explanation:

Step1: Recall the section formula

The section formula for a point \( B(x,y) \) dividing the line segment joining \( A(x_1,y_1) \) and \( C(x_2,y_2) \) 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, \( A(-3,4) \), \( C(2,-6) \), and the ratio \( AB:AC = 4:5 \), so \( m = 4 \), \( n=5 - 4=1 \) (since \( AB:BC=4:1 \) as \( AB + BC=AC \)).

Step2: Calculate the x - coordinate of B

Substitute \( x_1=-3 \), \( x_2 = 2 \), \( m = 4 \), \( n = 1 \) into the x - coordinate formula:

$$ x=\frac{4\times2+1\times(-3)}{4 + 1}=\frac{8-3}{5}=\frac{5}{5}=1 $$

Step3: Calculate the y - coordinate of B

Substitute \( y_1 = 4 \), \( y_2=-6 \), \( m = 4 \), \( n = 1 \) into the y - coordinate formula:

$$ y=\frac{4\times(-6)+1\times4}{4 + 1}=\frac{-24 + 4}{5}=\frac{-20}{5}=-4 $$

Answer:

\( (1, - 4) \)