Find the coordinates of point P along the directed line segment AB from A(-2,-8) to B(3,-3) so that ratio a AP to PB is 3 to 2

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.

Learn how to find the coordinates of point P on the line segment AB from A(-2, -8) to B(3, -3) with a ratio of 3:2 using the section formula for internal division. Detailed step-by-step solution provided.

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