Final answer:
To find the coordinates of point P along line segment AB with a ratio of AP to PB of 4 to 1, apply the section formula. Point P's coordinates are found to be (6.6, 3.8).
Step-by-step explanation:
To find the coordinates of point P along the directed line segment AB such that AP to PB is in the ratio 4 to 1, we use the section formula for internal division of a line segment. This formula gives us the coordinates of a point P(x, y) which divides a line segment connecting A(x1, y1) and B(x2, y2) in a given ratio m:n. In this case, A(1, 3), B(8, 4), and the ratio AP:PB is 4:1.
The formula for the x-coordinate of point P is:
x = (mx2 + nx1) / (m + n)
Substitute the given values:
x = (4 × 8 + 1 × 1) / (4 + 1)
x = (32 + 1) / 5
x = 33 / 5
x = 6.6
Similarly, the formula for the y-coordinate of point P is:
y = (my2 + ny1) / (m + n)
Substitute the given values:
y = (4 × 4 + 1 × 3) / (4 + 1)
y = (16 + 3) / 5
y = 19 / 5
y = 3.8
Therefore, the coordinates of point P are (6.6, 3.8).