To find point [tex]\( R \)[/tex] on the line segment [tex]\( P Q \)[/tex] that partitions it into the segments [tex]\( P R \)[/tex] and [tex]\( R Q \)[/tex] in the ratio 3:2, we can use the section formula. Specifically, for two points [tex]\( P(x_1, y_1) \)[/tex] and [tex]\( Q(x_2, y_2) \)[/tex] and a ratio [tex]\( m:n \)[/tex], the coordinates of point [tex]\( R(x_R, y_R) \)[/tex] that divides the segment [tex]\( P Q \)[/tex] internally in the ratio [tex]\( m:n \)[/tex] are given by:
[tex]\[
x_R = \frac{m x_2 + n x_1}{m + n}
\][/tex]
[tex]\[
y_R = \frac{m y_2 + n y_1}{m + n}
\][/tex]
Here, point [tex]\( P \)[/tex] has coordinates [tex]\( (-1, 3) \)[/tex], point [tex]\( Q \)[/tex] has coordinates [tex]\( (4, -2) \)[/tex], and the ratio [tex]\( m:n \)[/tex] is 3:2.
First, let's determine the [tex]\( x \)[/tex]-coordinate [tex]\( x_R \)[/tex]:
[tex]\[
x_R = \frac{3 \cdot 4 + 2 \cdot (-1)}{3+2}
\][/tex]
[tex]\[
x_R = \frac{12 - 2}{5}
\][/tex]
[tex]\[
x_R = \frac{10}{5}
\][/tex]
[tex]\[
x_R = 2
\][/tex]
Next, let's determine the [tex]\( y \)[/tex]-coordinate [tex]\( y_R \)[/tex]:
[tex]\[
y_R = \frac{3 \cdot (-2) + 2 \cdot 3}{3+2}
\][/tex]
[tex]\[
y_R = \frac{-6 + 6}{5}
\][/tex]
[tex]\[
y_R = \frac{0}{5}
\][/tex]
[tex]\[
y_R = 0
\][/tex]
Therefore, the coordinates of point [tex]\( R \)[/tex] are [tex]\( (2.0, 0.0) \)[/tex].
Since none of the provided answer choices match the coordinates [tex]\( (2.0, 0.0) \)[/tex], the correct response must be absent from the given options. However, based on our calculation, the correct coordinates of point [tex]\( R \)[/tex] that partitions [tex]\( P Q \)[/tex] into the ratio 3:2 are:
[tex]\[
\boxed{(2.0, 0.0)}
\][/tex]
