To find the coordinates of point $P(x, y)$ along the directed line segment $AB$ from $A(-2, -8)$ to $B(3, -3)$, such that the ratio $\frac{AP}{PB}$ is $\frac{3}{2}$, we can use the section formula for internal division.

The section formula for internal division is given by:

$P(x, y) = \left(\frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n}\right)$

Where:

• $(x_1, y_1)$ are the coordinates of point $A$
• $(x_2, y_2)$ are the coordinates of point $B$
• $m$ and $n$ are the given ratios

Given:

• $A(-2, -8)$
• $B(3, -3)$
• $\frac{AP}{PB} = \frac{3}{2}$, so $m = 3$ and $n = 2$

Substituting the values into the formula:

$P(x, y) = \left(\frac{3 \cdot 3 + 2 \cdot (-2)}{3+2}, \frac{3 \cdot (-3) + 2 \cdot (-8)}{3+2}\right)$

Calculating the coordinates:

$x = \frac{9 - 4}{5} = \frac{5}{5} = 1$ $y = \frac{-9 - 16}{5} = \frac{-25}{5} = -5$

So, the coordinates of point $P$ are $(1, -5)$.

Would you like more details or have any questions? Here are five related questions:

1. How would the coordinates change if the ratio was 2:3 instead of 3:2?
2. Can you find the length of segment $AP$ and $PB$ using the distance formula?
3. What if the ratio $\frac{AP}{PB}$ was 1:1? Where would $P$ be located?
4. How do you calculate the midpoint of a line segment? What is its significance?
5. What is the significance of the section formula in vector geometry?

Tip: In problems involving ratios and line segments, using the section formula is a quick and efficient way to find precise coordinates of a point dividing the segment in a given ratio.