To find the coordinates of point X that partitions segment AB in the ratio AX:BX = 1:2, we can use the section formula. According to the section formula, if point X divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, the coordinates of point X are given by:
\[ X\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
In this case, we have:
- A = (-3, 1)
- B = (-8, 5)
- The ratio m:n = 1:2 (so m = 1, n = 2)
Using this information, we can calculate the coordinates of point X:
- Calculate the x-coordinate of X:
\[ X_x = \frac{1 \cdot (-8) + 2 \cdot (-3)}{1 + 2} = \frac{-8 - 6}{3} = \frac{-14}{3} = -\frac{14}{3} \]
- Calculate the y-coordinate of X:
\[ X_y = \frac{1 \cdot 5 + 2 \cdot 1}{1 + 2} = \frac{5 + 2}{3} = \frac{7}{3} \]
Thus, the coordinates of point X are:
\[ X\left(-\frac{14}{3}, \frac{7}{3}\right) \]
However, let's evaluate the corresponding fractions given in the options to see if one of them matches.
- \(-\frac{14}{3}\) is equal to \(-4.\overline{6}\) or approximately -4.67.
- \(\frac{7}{3}\) is equal to approximately 2.33.
Now from the options listed:
- \(X\left(-\frac{31}{3}, 5\right)\)
- \(X\left(-\frac{10}{3}, \frac{8}{3}\right)\)
- \(X\left(-\frac{13}{3}, -10\right)\)
- \(X\left(-\frac{4}{3}, 2.13\right)\)
None of the options given in your list correspond to the ratio-derived coordinates of \(X\). Please check the possible answers or the calculations to find the correct option. Based on the calculation, the answer should logically be:
\[ X\left(-\frac{14}{3}, \frac{7}{3}\right) \]
This value seems absent from the options provided.