5. Point \( P \) divides the directed line segment from point \( A(-4,-1) \) to point \( B(6,4) \) in the ratio \( 2: 3 \). What are the coordinates of point \( P \) ?

To find the coordinates of point \( P \) that divides the line segment from point \( A(-4,-1) \) to point \( B(6,4) \) in the ratio \( 2:3 \), we can use the section formula. The coordinates of point \( P \) can be calculated using the formula: \[ P(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n} \right) \] where \( m \) and \( n \) are the parts of the ratio and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points \( A \) and \( B \) respectively. Here, \( m = 2 \) and \( n = 3 \): - \( A(-4, -1) \) gives \( x_1 = -4 \) and \( y_1 = -1 \) - \( B(6, 4) \) gives \( x_2 = 6 \) and \( y_2 = 4 \) Now substituting the values: \[ P(x) = \frac{2 \cdot 6 + 3 \cdot (-4)}{2 + 3} = \frac{12 - 12}{5} = \frac{0}{5} = 0 \] \[ P(y) = \frac{2 \cdot 4 + 3 \cdot (-1)}{2 + 3} = \frac{8 - 3}{5} = \frac{5}{5} = 1 \] Thus, the coordinates of point \( P \) are \( (0, 1) \).