To solve this problem, let's go through the process step by step:
1. Find the Coordinates of Point C:
Point [tex]\( C \)[/tex] divides the line segment [tex]\( AB \)[/tex] in the ratio 2:1. We can use the section formula to find the coordinates of [tex]\( C \)[/tex].
Given:
[tex]\( A(2, 3) \)[/tex]
[tex]\( B(-4, 1) \)[/tex]
Ratio [tex]\( = 2:1 \)[/tex]
Using the section formula, the coordinates of [tex]\( C \)[/tex] can be calculated as:
[tex]\[
C \left(\frac{{m x_2 + n x_1}}{{m + n}}, \frac{{m y_2 + n y_1}}{{m + n}} \right)
\][/tex]
Here, [tex]\( m = 2 \)[/tex], [tex]\( n = 1 \)[/tex], [tex]\( x_1 = 2 \)[/tex], [tex]\( y_1 = 3 \)[/tex], [tex]\( x_2 = -4 \)[/tex], and [tex]\( y_2 = 1 \)[/tex].
[tex]\[
C_x = \frac{{2 \cdot (-4) + 1 \cdot 2}}{{2 + 1}} = \frac{{-8 + 2}}{{3}} = \frac{{-6}}{{3}} = -2
\][/tex]
[tex]\[
C_y = \frac{{2 \cdot 1 + 1 \cdot 3}}{{2 + 1}} = \frac{{2 + 3}}{{3}} = \frac{{5}}{{3}} \approx 1.6667
\][/tex]
So, the coordinates of point [tex]\( C \)[/tex] are [tex]\( (-2, 1.6667) \)[/tex].
2. Find the Slope of Line AB:
The slope (m) of line segment [tex]\( AB \)[/tex] can be found using the formula:
[tex]\[
m_{AB} = \frac{{y_2 - y_1}}{{x_2 - x_1}}
\][/tex]
[tex]\[
m_{AB} = \frac{{1 - 3}}{{-4 - 2}} = \frac{{-2}}{{-6}} = \frac{{1}}{{3}}
\][/tex]
So, the slope of [tex]\( AB \)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
3. Find the Slope of the Perpendicular Line:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So,
[tex]\[
m_{perpendicular} = -\frac{1}{m_{AB}} = -3
\][/tex]
The slope of the line perpendicular to [tex]\( AB \)[/tex] is [tex]\( -3 \)[/tex].
4. Find the Equation of the Perpendicular Line Passing Through Point C:
The equation of a line in slope-intercept form (y = mx + b) can be written as:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\( m = -3 \)[/tex], and the line passes through [tex]\( C(-2, 1.6667) \)[/tex].
Substituting these values:
[tex]\[
y - 1.6667 = -3(x + 2)
\][/tex]
[tex]\[
y - 1.6667 = -3x - 6
\][/tex]
[tex]\[
y = -3x - 6 + 1.6667
\][/tex]
[tex]\[
y = -3x - 4.3333
\][/tex]
Hence, the equation of the line that passes through the point [tex]\( C \)[/tex] and is perpendicular to [tex]\( AB \)[/tex] is:
[tex]\[
y = -3x - 4.3333
\][/tex]
This is the required equation.
