To find the coordinates of point \( X \) on line segment \( AB \) such that the ratio of \( AX \) to \( XB \) is \( 3:5 \), we can use the section formula. The coordinates of points \( A \) and \( B \) are given as follows:
- \( A(-1, 2) \)
- \( B(3, 6) \)
Since the ratio \( AX:XB = 3:5 \), we can express it as \( m:n = 3:5 \), where \( m = 3 \) and \( n = 5 \).
Using the section formula, point \( X \) that divides the line segment \( AB \) in the ratio \( m:n \) can be determined by the following formula:
\[ X = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right) \]
Substituting the values:
- \( x_1 = -1 \), \( y_1 = 2 \) (coordinates of A)
- \( x_2 = 3 \), \( y_2 = 6 \) (coordinates of B)
We have:
\[ X = \left( \frac{3 \cdot 3 + 5 \cdot (-1)}{3+5}, \frac{3 \cdot 6 + 5 \cdot 2}{3+5} \right) \]
Calculating the x-coordinate:
\[ X_x = \frac{9 - 5}{8} = \frac{4}{8} = \frac{1}{2} \]
Calculating the y-coordinate:
\[ X_y = \frac{18 + 10}{8} = \frac{28}{8} = \frac{7}{2} \]
Thus, the coordinates of point \( X \) are \( \left( \frac{1}{2}, \frac{7}{2} \right) \).
Now let's convert \( \frac{7}{2} \) to a mixed number:
\[ \frac{7}{2} = 3 \frac{1}{2} \]
So, the coordinates \( X \) can be written as \( (0.5, 3.5) \) or \( \left( \frac{1}{2}, 3 \frac{1}{2} \right) \).
Therefore, the answer is:
B. \( \left( \frac{1}{2}, 3 \frac{1}{2} \right) \).