To find the point [tex]\( R \)[/tex] that partitions the line segment [tex]\( PQ \)[/tex] with coordinates [tex]\( P(6, -5) \)[/tex] and [tex]\( Q(-2, 4) \)[/tex] in the ratio [tex]\( 3:2 \)[/tex], we need to use the section formula.
The section formula for a point [tex]\( R(x, y) \)[/tex] that divides the line segment joining [tex]\( P(x_1, y_1) \)[/tex] and [tex]\( Q(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ R_x = \frac{mx_2 + nx_1}{m + n} \][/tex]
[tex]\[ R_y = \frac{my_2 + ny_1}{m + n} \][/tex]
Here, point [tex]\( P \)[/tex] is [tex]\( (x_1, y_1) = (6, -5) \)[/tex] and point [tex]\( Q \)[/tex] is [tex]\( (x_2, y_2) = (-2, 4) \)[/tex]. The ratio [tex]\( m:n = 3:2 \)[/tex].
We substitute these values into the section formula:
1. Calculate the [tex]\( x \)[/tex]-coordinate of point [tex]\( R \)[/tex]:
[tex]\[ R_x = \frac{3(-2) + 2(6)}{3 + 2} \][/tex]
[tex]\[ R_x = \frac{-6 + 12}{5} \][/tex]
[tex]\[ R_x = \frac{6}{5} \][/tex]
[tex]\[ R_x = 1.2 \][/tex]
2. Calculate the [tex]\( y \)[/tex]-coordinate of point [tex]\( R \)[/tex]:
[tex]\[ R_y = \frac{3(4) + 2(-5)}{3 + 2} \][/tex]
[tex]\[ R_y = \frac{12 - 10}{5} \][/tex]
[tex]\[ R_y = \frac{2}{5} \][/tex]
[tex]\[ R_y = 0.4 \][/tex]
So, the coordinates of point [tex]\( R \)[/tex] are [tex]\( (1.2, 0.4) \)[/tex].
By matching these coordinates with the given options, we find that the correct answer is:
[tex]\[ \boxed{\left(\frac{6}{5}, \frac{2}{5}\right)} \][/tex]
None of the options directly match [tex]\( (1.2, 0.4) \)[/tex], but this corresponds to:
[tex]\[ \frac{6}{5} = 1.2 \quad \text{and} \quad \frac{2}{5} = 0.4 \][/tex]
The option [tex]\( D \)[/tex] appears almost similar to the calculated values. However, the x-coordinate has a typo in option D which should be corrected to [tex]\(\frac{6}{5}\)[/tex].
Hence, the closet and appropriate option from provided choices should be corrected as:
[tex]\[ D. \left(\frac{6}{5}, \frac{2}{5}\right) \][/tex] considering intended calculation mechanism behind the solution.
