Final answer:
The coordinates of point p along the directed line segment ab so that ap to pb is the given ratio are (5, -0.2).
Step-by-step explanation:
To find the coordinates of point P along the directed line segment from A(3,1) to B(8, -2) with the ratio of AP to PB being 2 to 3, we can use the section formula. This formula gives us the coordinates of a point that divides a segment into a certain ratio. The section formula for a ratio m:n is given by:
P(x, y) = ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n))
For the points A and B with given coordinates and the ratio 2:3, we substitute into the formula as follows:
- x1 = 3
- y1 = 1
- x2 = 8
- y2 = -2
- m = 2
- n = 3
Substitute these values into the section formula:
P(x, y) = ((2*8 + 3*3) / (2 + 3), (2*-2 + 3*1) / (2 + 3))
P(x, y) = ((16 + 9) / 5, (-4 + 3) / 5)
P(x, y) = (25 / 5, -1 / 5)
P(x, y) = (5, -0.2)
Therefore, the coordinates of point P are (5, -0.2).