To find the coordinates of point [tex]\( R \)[/tex] that partitions the line segment [tex]\( PQ \)[/tex] in the ratio [tex]\( 3:2 \)[/tex], we will use the section formula. The section formula states that if a point [tex]\( R \)[/tex] divides a line segment [tex]\( PQ \)[/tex] with coordinates [tex]\( P(x_1, y_1) \)[/tex] and [tex]\( Q(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( R \)[/tex] are given by:
[tex]\[
R\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
We are given:
- [tex]\( P(6, -5) \)[/tex]
- [tex]\( Q(-2, 4) \)[/tex]
- Ratio [tex]\( m:n = 3:2 \)[/tex]
Plugging the given values into the section formula, we calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( R \)[/tex]:
[tex]\[
x_R = \frac{3(-2) + 2(6)}{3 + 2} = \frac{(-6) + 12}{5} = \frac{6}{5}
\][/tex]
Next, we calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( R \)[/tex]:
[tex]\[
y_R = \frac{3(4) + 2(-5)}{3 + 2} = \frac{12 + (-10)}{5} = \frac{2}{5}
\][/tex]
Therefore, the coordinates of point [tex]\( R \)[/tex] are:
[tex]\[
R\left( \frac{6}{5}, \frac{2}{5} \right)
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\left( \frac{6}{5}, \frac{2}{5} \right)}
\][/tex]
