Find the coordinates of point [tex] P [/tex] along the directed line segment [tex] AB [/tex] from [tex] A(-2, -8) [/tex] to [tex] B(3, -3) [/tex] such that the ratio of [tex] AP [/tex] to [tex] PB [/tex] is 3 to 2.

Given points:
- [tex] A(-2, -8) [/tex]
- [tex] B(3, -3) [/tex]

Ratio:
- [tex] AP : PB = 3 : 2 [/tex]

Calculate the coordinates of [tex] P [/tex].

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].