To find the point [tex]\( R \)[/tex] that partitions the line segment [tex]\( PQ \)[/tex] in the ratio [tex]\( 3:2 \)[/tex], we use the section formula. The section formula for internal division states that if a point [tex]\( R \)[/tex] divides the line segment joining two points [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]\[
R(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Here, the coordinates of point [tex]\( P \)[/tex] are [tex]\( (6, -5) \)[/tex] and those of point [tex]\( Q \)[/tex] are [tex]\( (-2, 4) \)[/tex]. The ratio given is [tex]\( 3:2 \)[/tex].
First, let's find the [tex]\( x \)[/tex]-coordinate of point [tex]\( R \)[/tex]:
[tex]\[
R_x = \frac{m Q_x + n P_x}{m+n} = \frac{3(-2) + 2(6)}{3+2} = \frac{(-6) + 12}{5} = \frac{6}{5} = 1.2
\][/tex]
Next, let's find the [tex]\( y \)[/tex]-coordinate of point [tex]\( R \)[/tex]:
[tex]\[
R_y = \frac{m Q_y + n P_y}{m+n} = \frac{3(4) + 2(-5)}{3+2} = \frac{12 + (-10)}{5} = \frac{2}{5} = 0.4
\][/tex]
Therefore, the coordinates of point [tex]\( R \)[/tex] are [tex]\( (1.2, 0.4) \)[/tex].
Hence, the correct answer is [tex]\( \boxed{\left(\frac{6}{5}, \frac{2}{5}\right)} \)[/tex], which corresponds to option A.
