To determine the coordinates of point [tex]\( M \)[/tex], which partitions the directed line segment from [tex]\( L(-6,2) \)[/tex] to [tex]\( N(5,-3) \)[/tex] in the ratio [tex]\( 2:5 \)[/tex], we use the section formula for internal division. The section formula states that the coordinates of a point dividing the line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex] are given by:
[tex]\[
x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\][/tex]
[tex]\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\][/tex]
In this case, the coordinates of [tex]\( L \)[/tex] are [tex]\((x_1, y_1) = (-6, 2)\)[/tex] and the coordinates of [tex]\( N \)[/tex] are [tex]\((x_2, y_2) = (5, -3)\)[/tex]. The ratio [tex]\( m:n \)[/tex] is [tex]\( 2:5 \)[/tex].
Now let's calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( M \)[/tex]:
[tex]\[
x = \frac{2 \cdot 5 + 5 \cdot (-6)}{2 + 5}
\][/tex]
[tex]\[
x = \frac{10 - 30}{7}
\][/tex]
[tex]\[
x = \frac{-20}{7}
\][/tex]
[tex]\[
x = -2.857142857142857
\][/tex]
Next, let's calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( M \)[/tex]:
[tex]\[
y = \frac{2 \cdot (-3) + 5 \cdot 2}{2 + 5}
\][/tex]
[tex]\[
y = \frac{-6 + 10}{7}
\][/tex]
[tex]\[
y = \frac{4}{7}
\][/tex]
[tex]\[
y = 0.5714285714285714
\][/tex]
Thus, the coordinates of point [tex]\( M \)[/tex] that partitions the segment [tex]\( LN \)[/tex] in the ratio [tex]\( 2:5 \)[/tex] are:
[tex]\[
x = -2.857142857142857 \checkmark
\][/tex]
[tex]\[
y = 0.5714285714285714
\][/tex]
