To find the coordinates of the point [tex]\( R \)[/tex] that divides 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 divides a line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of the point are given by:
[tex]\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right).
\][/tex]
Given:
- [tex]\( P(6, -5) \)[/tex]
- [tex]\( Q(-2, 4) \)[/tex]
- Ratio [tex]\( m:n = 3:2 \)[/tex]
We will now apply the section formula step-by-step:
1. Identify [tex]\( x_1, y_1, x_2, y_2, m, \)[/tex] and [tex]\( n \)[/tex]:
- [tex]\( P_x = 6 \)[/tex]
- [tex]\( P_y = -5 \)[/tex]
- [tex]\( Q_x = -2 \)[/tex]
- [tex]\( Q_y = 4 \)[/tex]
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 2 \)[/tex]
2. Calculate the x-coordinate of [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]
3. Calculate the y-coordinate of [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]
So, the coordinates of point [tex]\( R \)[/tex] are [tex]\( (1.2, 0.4) \)[/tex].
Based on the given options, none of the coordinates match [tex]\((1.2, 0.4)\)[/tex] directly. However, we have found the correct partitioning point through calculations and the answer is confirmed to be [tex]\( (1.2, 0.4) \)[/tex].
