given AB with A(8,-4) and b(-6,-11), if p partitions AB such tht the ratio of AP to  AB is 2:7, find the coordinates of P

can you solve this but with this formula: k(x2-x1)+x1
k(y2-y1)+y1

To find the coordinates of point \( P(x, y) \) that partitions the line segment \( AB \) in the ratio \( 2:7 \), we can use the given formula for the coordinates:

\[
P(x, y) = \left( \frac{k(x_2 - x_1)}{m+n} + x_1, \frac{k(y_2 - y_1)}{m+n} + y_1 \right)
\]

Given:
- \( A(8, -4) \)
- \( B(-6, -11) \)
- Ratio \( AP : PB = 2 : 7 \), so \( m = 2 \) and \( n = 7 \).

**Coordinates of \( P \):**

1. Calculate the x-coordinate:

\[
x = \frac{k(x_2 - x_1)}{m+n} + x_1
\]

Here, \( x_1 = 8 \), \( x_2 = -6 \), and \( k = \frac{2}{2+7} = \frac{2}{9} \).

\[
x = \frac{2(-6 - 8)}{9} + 8 = \frac{2(-14)}{9} + 8 = \frac{-28}{9} + 8 = \frac{-28}{9} + \frac{72}{9} = \frac{44}{9}
\]

2. Calculate the y-coordinate:

\[
y = \frac{k(y_2 - y_1)}{m+n} + y_1
\]

Here, \( y_1 = -4 \), \( y_2 = -11 \), and \( k = \frac{2}{9} \).

\[
y = \frac{2(-11 - (-4))}{9} + (-4) = \frac{2(-7)}{9} - 4 = \frac{-14}{9} - \frac{36}{9} = \frac{-50}{9}
\]

Thus, the coordinates of point \( P \) are:

\[
P\left(\frac{44}{9}, -\frac{50}{9}\right)
\]

Would you like any further details on how this calculation was done?

Here are five related questions:
1. How would the coordinates of \( P \) change if the ratio was \( 3:4 \) instead?
2. Can you find the midpoint of \( AB \) and compare it with point \( P \)?
3. How does changing the ratio affect the location of \( P \) relative to \( A \) and \( B \)?
4. What would the coordinates of \( P \) be if the ratio was reversed, i.e., \( 7:2 \)?
5. How would you determine \( P \)'s coordinates if the ratio \( AP:PB \) was given as a decimal instead of a fraction?

**Tip:** Understanding the concept of section formulas can help you solve a wide range of problems involving dividing lines in a specific ratio.

Learn how to find the coordinates of point P that divides line segment AB in the ratio 2:7 using the section formula. Detailed step-by-step solution with coordinates calculation for points A(8, -4) and B(-6, -11). Suitable for high school students studying coordinate geometry.

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