Final answer:
To find the coordinates of point P along directed line segment AB with the given conditions, we apply the section formula. However, after calculating and rounding, the correct answer does not match any of the provided choices, indicating a possible mistake in the options.
Step-by-step explanation:
The question asks us to find the coordinates of point P along directed line segment AB, with A(4,8) and B(16, -2), such that the ratio of AP to PB is 3 to 2. We can solve this by using the concept of section formula in coordinate geometry which is used to find a point that divides a line segment in a given ratio.
Since the ratio of AP to PB is 3:2, we can use the section formula to calculate the coordinates of P. We apply the formula for internal division:
P(x, y) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)),
where m:n is the ratio, and (x1, y1) & (x2, y2) are the coordinates of A and B respectively. Substituting the given values, we get:
P(x, y) = ((2*16 + 3*4)/(2+3), (2*(-2) + 3*8)/(2+3))
= ((32 + 12)/5, (-4 + 24)/5)
= (44/5, 20/5)
= (8.8, 4)However, none of the answer choices exactly match the coordinates we found, so we re-examine the calculation and options given. The correct coordinates of P, (x,y), must be rounded to the nearest whole number if the exact value is not provided as an option. After rounding, we get (9,4), but this is still not among the choices. Therefore, it seems there may be a mistake in the given answer choices or a misinterpretation in the calculation.