To find the coordinates of point \( X \) on line segment \( \overline{AB} \) such that the ratio of \( AX \) to \( XB \) is \( 3:5 \), we first need to determine the position of point \( X \) along the line segment defined by points \( A(-1, 2) \) and \( B(3, 6) \).
Using the section formula, we can find the coordinates of point \( X \) dividing \( AB \) in the ratio \( m:n = 3:5 \). The coordinates \( (x, y) \) of point \( X \) can be calculated as follows:
\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m+n} \] \[ y = \frac{m \cdot y_2 + n \cdot y_1}{m+n} \]
Where:
- \( A(x_1, y_1) = (-1, 2) \)
- \( B(x_2, y_2) = (3, 6) \)
- \( m = 3 \) (part of ratio towards point \( B \))
- \( n = 5 \) (part of ratio towards point \( A \))
Substituting the values:
For x-coordinate: \[ x = \frac{3 \cdot 3 + 5 \cdot (-1)}{3 + 5} = \frac{9 - 5}{8} = \frac{4}{8} = \frac{1}{2} \]
For y-coordinate: \[ y = \frac{3 \cdot 6 + 5 \cdot 2}{3 + 5} = \frac{18 + 10}{8} = \frac{28}{8} = \frac{7}{2} \]
Thus, the coordinates of point \( X \) are: \[ X\left(\frac{1}{2}, \frac{7}{2}\right) \]
Now, we compare these coordinates to the given responses:
- (−14,312)
- (12,312)
- (112,112)
- (125,425)
None of the given options directly match the calculated coordinates, \( \left(\frac{1}{2}, \frac{7}{2}\right) \).
The correct coordinates for point \( X \) dividing \( \overline{AB} \) in the ratio \( 3:5 \) are:
\[ \boxed{\left(\frac{1}{2}, \frac{7}{2}\right)} \]