Answer :
To find the coordinates of point [tex]\( P \)[/tex] along the directed line segment from [tex]\( A(-2, -8) \)[/tex] to [tex]\( B(3, -3) \)[/tex] with the ratio [tex]\( AP:PB = 3:2 \)[/tex], we can use the section formula. This formula allows us to determine the coordinates of a point dividing a line segment into a given ratio.
Here's a step-by-step guide for the solution:
1. Identify the coordinates and the ratio:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (-2, -8) \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( (3, -3) \)[/tex].
- The ratio in which point [tex]\( P \)[/tex] divides the line segment [tex]\( AB \)[/tex] is [tex]\( 3:2 \)[/tex].
2. Use the section formula:
The section formula for a point [tex]\( P(x, y) \)[/tex] dividing the line segment [tex]\( AB \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
x = \frac{n \cdot x_1 + m \cdot x_2}{m + n}
\][/tex]
[tex]\[
y = \frac{n \cdot y_1 + m \cdot y_2}{m + n}
\][/tex]
Plugging in the known values:
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 2 \)[/tex]
- [tex]\( x_1 = -2 \)[/tex], [tex]\( y_1 = -8 \)[/tex]
- [tex]\( x_2 = 3 \)[/tex], [tex]\( y_2 = -3 \)[/tex]
3. Calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[
x = \frac{2 \cdot (-2) + 3 \cdot 3}{3 + 2} = \frac{-4 + 9}{5} = \frac{5}{5} = 1.0
\][/tex]
4. Calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[
y = \frac{2 \cdot (-8) + 3 \cdot (-3)}{3 + 2} = \frac{-16 - 9}{5} = \frac{-25}{5} = -5.0
\][/tex]
Thus, the coordinates of point [tex]\( P \)[/tex] are [tex]\( (1.0, -5.0) \)[/tex].
Here's a step-by-step guide for the solution:
1. Identify the coordinates and the ratio:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (-2, -8) \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( (3, -3) \)[/tex].
- The ratio in which point [tex]\( P \)[/tex] divides the line segment [tex]\( AB \)[/tex] is [tex]\( 3:2 \)[/tex].
2. Use the section formula:
The section formula for a point [tex]\( P(x, y) \)[/tex] dividing the line segment [tex]\( AB \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
x = \frac{n \cdot x_1 + m \cdot x_2}{m + n}
\][/tex]
[tex]\[
y = \frac{n \cdot y_1 + m \cdot y_2}{m + n}
\][/tex]
Plugging in the known values:
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 2 \)[/tex]
- [tex]\( x_1 = -2 \)[/tex], [tex]\( y_1 = -8 \)[/tex]
- [tex]\( x_2 = 3 \)[/tex], [tex]\( y_2 = -3 \)[/tex]
3. Calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[
x = \frac{2 \cdot (-2) + 3 \cdot 3}{3 + 2} = \frac{-4 + 9}{5} = \frac{5}{5} = 1.0
\][/tex]
4. Calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( P \)[/tex]:
[tex]\[
y = \frac{2 \cdot (-8) + 3 \cdot (-3)}{3 + 2} = \frac{-16 - 9}{5} = \frac{-25}{5} = -5.0
\][/tex]
Thus, the coordinates of point [tex]\( P \)[/tex] are [tex]\( (1.0, -5.0) \)[/tex].
