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.