To find the coordinates of point P on directed line segment AB that partitions AB so that AP: AB is 3:5, we can use the formula for finding the point that lies a certain fraction of the way between two other points:
P = (1 - t)A + tB
where A and B are the coordinates of the two points, and t is a value between 0 and 1 that determines where the point P lies on the line segment AB.
In this case, we are told that the ratio of AP to AB is 3:5. We can set t equal to 3/5, which means that point P is 3/5 of the way from A to B along the line segment AB:
P = (1 - 3/5)A + (3/5)B
= (2/5)A + (3/5)B
Substituting the coordinates of A and B, we get:
P = (2/5)(-4, -2) + (3/5)(6, 3)
= (-8/5, -6/5) + (18/5, 9/5)
= (10/5, 3/5)
= (2, 3/5)
Therefore, the coordinates of point P are (2, 3/5).