Given:
- Point \(P\) at coordinates \((-10, 3)\).
- Point \(R\) at coordinates \((4, 7)\).
- Ratio \(P R: R Q = 2:3\).
To find the coordinates of point \(Q\), we use the section formula for a point dividing a line segment internally in a given ratio. The section formula for a point dividing the segment joining \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m:n\) is:
[tex]\[
(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Given that point \(R\) divides segment \(P Q\) in the ratio \(2:3\), we need to essentially reverse-engineer the section formula to determine the coordinates of \(Q\).
Let's denote the coordinates of \(Q\) as \((x_Q, y_Q)\).
Using the coordinates of \(R\) and the ratio, the section formula gives the coordinates of \(R\):
[tex]\[
R_x = \frac{2 \cdot x_Q + 3 \cdot (-10)}{5}
\][/tex]
Given \(R_x = 4\), we can set up the equation:
[tex]\[
4 = \frac{2 x_Q - 30}{5}
\][/tex]
Multiplying both sides by 5:
[tex]\[
20 = 2 x_Q - 30
\][/tex]
Adding 30 to both sides:
[tex]\[
50 = 2 x_Q
\][/tex]
Dividing by 2:
[tex]\[
x_Q = 25
\][/tex]
Similarly, for the y-coordinate:
[tex]\[
R_y = \frac{2 \cdot y_Q + 3 \cdot 3}{5}
\][/tex]
Given \(R_y = 7\), we can set up the equation:
[tex]\[
7 = \frac{2 y_Q + 9}{5}
\][/tex]
Multiplying both sides by 5:
[tex]\[
35 = 2 y_Q + 9
\][/tex]
Subtracting 9 from both sides:
[tex]\[
26 = 2 y_Q
\][/tex]
Dividing by 2:
[tex]\[
y_Q = 13
\][/tex]
Therefore, the coordinates of point \(Q\) are \((25, 13)\).
The correct answer is:
[tex]\[ B. (25, 13) \][/tex]
