The directed line segment from [tex]\( L \)[/tex] to [tex]\( N \)[/tex] has endpoints [tex]\( L(-6, 2) \)[/tex] and [tex]\( N(5, -3) \)[/tex]. What are the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( M \)[/tex], which partitions the directed line segment into the ratio 2:5?

[tex]\[ x = \][/tex]
[tex]\[ y = \][/tex]



Answer :

To find the coordinates of point M that partitions the line segment from point L to point N in the given ratio of 2:5, we use the section formula. The section formula for dividing a line segment in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ M = \left( \frac{n x_1 + m x_2}{m + n}, \frac{n y_1 + m y_2}{m + n} \right) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] are the coordinates of [tex]\( L \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of [tex]\( N \)[/tex].

Given:
- Coordinates of [tex]\( L \)[/tex] are [tex]\( (-6, 2) \)[/tex]
- Coordinates of [tex]\( N \)[/tex] are [tex]\( (5, -3) \)[/tex]
- Ratio [tex]\( m:n \)[/tex] is [tex]\( 2:5 \)[/tex]

Let [tex]\( m = 2 \)[/tex] and [tex]\( n = 5 \)[/tex].

We now substitute these values into the section formula.

For the x-coordinate of point M:
[tex]\[ x_M = \frac{n \cdot x_L + m \cdot x_N}{m + n} = \frac{5 \cdot (-6) + 2 \cdot 5}{2 + 5} \][/tex]

Calculation:
[tex]\[ x_M = \frac{(5 \cdot -6) + (2 \cdot 5)}{7} = \frac{-30 + 10}{7} = \frac{-20}{7} \approx -2.8571 \][/tex]

For the y-coordinate of point M:
[tex]\[ y_M = \frac{n \cdot y_L + m \cdot y_N}{m + n} = \frac{5 \cdot 2 + 2 \cdot (-3)}{2 + 5} \][/tex]

Calculation:
[tex]\[ y_M = \frac{(5 \cdot 2) + (2 \cdot -3)}{7} = \frac{10 - 6}{7} = \frac{4}{7} \approx 0.5714 \][/tex]

Thus, the coordinates of point M are:
[tex]\[ (x, y) = (-2.8571, 0.5714) \][/tex]

Therefore, the x-coordinate of point [tex]\( M \)[/tex] is [tex]\(-2.8571\)[/tex] and the y-coordinate of point [tex]\( M \)[/tex] is [tex]\( 0.5714 \)[/tex].