To find the coordinates of point [tex]\( S \)[/tex] that partitions the segment from [tex]\( R(-14, -1) \)[/tex] to [tex]\( T(4, -13) \)[/tex] in the ratio of [tex]\( 1:5 \)[/tex], we use the section formula for dividing a line segment internally.
The section formula states that if a point [tex]\( S \)[/tex] divides the segment joining [tex]\( R(x_1, y_1) \)[/tex] and [tex]\( T(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( S \)[/tex], let's denote them by [tex]\( (x_S, y_S) \)[/tex], can be determined by the formula:
[tex]\[
x_S = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\][/tex]
[tex]\[
y_S = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\][/tex]
Here, we have:
- [tex]\( R(x_1, y_1) = (-14, -1) \)[/tex]
- [tex]\( T(x_2, y_2) = (4, -13) \)[/tex]
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 5 \)[/tex]
Substituting these values into the formulas, we get:
[tex]\[
x_S = \frac{1 \cdot 4 + 5 \cdot (-14)}{1 + 5}
\][/tex]
[tex]\[
y_S = \frac{1 \cdot (-13) + 5 \cdot (-1)}{1 + 5}
\][/tex]
Calculating [tex]\( x_S \)[/tex]:
[tex]\[
x_S = \frac{4 + 5 \cdot (-14)}{6} = \frac{4 - 70}{6} = \frac{-66}{6} = -11
\][/tex]
Calculating [tex]\( y_S \)[/tex]:
[tex]\[
y_S = \frac{-13 + 5 \cdot (-1)}{6} = \frac{-13 - 5}{6} = \frac{-18}{6} = -3
\][/tex]
Therefore, the coordinates of point [tex]\( S \)[/tex] are:
[tex]\[
(-11, -3)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{(-11, -3)}
\][/tex]
