To determine the coordinates of point P, which divides the line segment AB in the ratio 3:2, we can use the section formula. The section formula helps find the coordinates of a point that divides a line segment internally in a given ratio.
Given:
- Coordinates of point A: \(\( A(-6, -5) \)\)
- Coordinates of point B: \(\( B(4, 0) \)\)
- The ratio \(\( AP:PB = 3:2 \)\)
According to the section formula, if a point P divides the line segment joining points \(\( (x_1, y_1) \)\) and \(\( (x_2, y_2) \)\) in the ratio \(\( m:n \)\), then the coordinates of P, i.e., \(\( P(x, y) \)\) are given by:
\(\[
x = \frac{m x_2 + n x_1}{m + n}
\]\)
\(\[
y = \frac{m y_2 + n y_1}{m + n}
\]\)
Where:
- \(\( (x_1, y_1) \)\) are the coordinates of point A
- \(\( (x_2, y_2) \)\) are the coordinates of point B
- \(\( m \)\) and \(\( n \)\) are the given ratio parts
Substituting the given values:
- \(\( x_1 = -6 \)\)
- \(\( y_1 = -5 \)\)
- \(\( x_2 = 4 \)\)
- \(\( y_2 = 0 \)\)
- \(\( m = 3 \)\)
- \(\( n = 2 \)\)
For \(\( x \)\):
\(\[
x = \frac{3 \cdot 4 + 2 \cdot (-6)}{3 + 2} = \frac{12 - 12}{5} = \frac{0}{5} = 0
\]\)
For \(\( y \)\):
\(\[
y = \frac{3 \cdot 0 + 2 \cdot (-5)}{3 + 2} = \frac{0 - 10}{5} = \frac{-10}{5} = -2
\]\)
Therefore, the coordinates of point P are:
\(\[
P(0, -2)
\]\)
