To find point P on line segment AB such that the ratio of \( AP \) to \( PB \) is \( 1:3 \), we first need to determine the coordinates of points A and B.
Let’s assume point A has coordinates \((x_1, y_1)\) and point B has coordinates \((x_2, y_2)\).
We can express point P as a weighted average of points A and B based on the given ratio. If the ratio \( AP:PB \) is \( 1:3 \), it implies that \( AP \) is \(\frac{1}{1+3} = \frac{1}{4}\) of the way from A to B, and \( PB \) is \(\frac{3}{4}\) of the way from A to B.
Using the section formula, the coordinates of P can be calculated as:
\[ P = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
where \( m = 3 \) (the portion to B) and \( n = 1 \) (the portion to A).
Without specific coordinates for points A and B, we cannot complete the calculation. However, if we assume general points for A and B, we can derive point P.
For example, let's choose points A and B:
- A(0, 0)
- B(4, 4)
Then, applying the section formula:
- \( P_x = \frac{34 + 10}{3 + 1} = \frac{12 + 0}{4} = 3 \)
- \( P_y = \frac{34 + 10}{3 + 1} = \frac{12 + 0}{4} = 3 \)
Thus, \( P \) would be (3, 3).
However, since we need the answer among the provided options, and assuming any of them could be derived from specific points A and B, you should check which coordinates yield the desired ratio across varying assumptions about A and B.
The options provided, without the exact coordinates for A and B, leave us to deduce from the plausible choices:
- \( P(-1, -1) \)
- \( P\left(-\frac{5}{2}, -\frac{9}{2}\right) \)
- \( P\left(\frac{1}{2}, \frac{3}{2}\right) \)
- \( P\left(-\frac{14}{5}, -\frac{13}{5}\right) \)
Without more information, we can't definitively choose which point corresponds to the requested ratio, so additional context about the locations of points A and B would clarify the required answer from the list you've given.