To determine the coordinates of point [tex]\( M \)[/tex] that divides the directed line segment from [tex]\( L \)[/tex] to [tex]\( N \)[/tex] in the ratio 2:5, you can use the section formula for internal division.
The section formula is given by:
[tex]\[ \left(\frac{k \cdot x_2 + m \cdot x_1}{k + m}, \frac{k \cdot y_2 + m \cdot y_1}{k + m}\right) \][/tex]
Here, [tex]\( L \)[/tex] and [tex]\( N \)[/tex] are endpoints with coordinates:
[tex]\[ L(x_1, y_1) = (-6, 2) \][/tex]
[tex]\[ N(x_2, y_2) = (5, -3) \][/tex]
The ratio [tex]\( k:m = 2:5 \)[/tex], so [tex]\( k = 2 \)[/tex] and [tex]\( m = 5 \)[/tex].
Now let's calculate the coordinates of [tex]\( M \)[/tex]:
### Calculation for the [tex]\( x \)[/tex]-coordinate:
[tex]\[
M_x = \frac{k \cdot x_2 + m \cdot x_1}{k + m} = \frac{2 \cdot 5 + 5 \cdot (-6)}{2 + 5}
\][/tex]
[tex]\[
M_x = \frac{10 - 30}{7} = \frac{-20}{7} = -2.857142857142857
\][/tex]
### Calculation for the [tex]\( y \)[/tex]-coordinate:
[tex]\[
M_y = \frac{k \cdot y_2 + m \cdot y_1}{k + m} = \frac{2 \cdot (-3) + 5 \cdot 2}{2 + 5}
\][/tex]
[tex]\[
M_y = \frac{-6 + 10}{7} = \frac{4}{7} = 0.5714285714285714
\][/tex]
Therefore, the coordinates of point [tex]\( M \)[/tex] which partitions the directed line segment from [tex]\( L \)[/tex] to [tex]\( N \)[/tex] in the ratio 2:5 are:
[tex]\[
M \left( -2.857142857142857, 0.5714285714285714 \right)
\][/tex]
