Let's solve the problem step-by-step to find the coordinates of point Q.
1. Identify Given Data:
- Coordinates of Point [tex]\(P\)[/tex]: [tex]\(P(-10, 3)\)[/tex]
- Coordinates of Point [tex]\(R\)[/tex]: [tex]\(R(4, 7)\)[/tex]
- Ratio [tex]\(PR: RQ = 2:3\)[/tex]
2. Section Formula:
The section formula helps us find the coordinates of a point dividing a segment in a given ratio. If a point [tex]\(R(x, y)\)[/tex] divides a 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], the coordinates [tex]\((x, y)\)[/tex] 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]
Here, [tex]\(x_1 = -10\)[/tex], [tex]\(y_1 = 3\)[/tex], [tex]\(x_2 = x\)[/tex] (coordinates of [tex]\(Q\)[/tex]), [tex]\(y_2 = y\)[/tex] (coordinates of [tex]\(Q\)[/tex]), [tex]\(m = 2\)[/tex], [tex]\(n = 3\)[/tex], and [tex]\(R = (4, 7)\)[/tex].
3. Setup Equations:
Using the section formula, write two separate equations for [tex]\(x\)[/tex]-coordinate and [tex]\(y\)[/tex]-coordinate using point [tex]\(R\)[/tex]'s known coordinates.
For the [tex]\(x\)[/tex]-coordinate:
[tex]\[
4 = \frac{2x + 3(-10)}{2 + 3}
\][/tex]
For the [tex]\(y\)[/tex]-coordinate:
[tex]\[
7 = \frac{2y + 3(3)}{2 + 3}
\][/tex]
4. Solve the [tex]\(x\)[/tex]-coordinate Equation:
[tex]\[
4 = \frac{2x - 30}{5}
\][/tex]
Multiply both sides by 5:
[tex]\[
20 = 2x - 30
\][/tex]
Add 30 to both sides:
[tex]\[
50 = 2x
\][/tex]
Divide both sides by 2:
[tex]\[
x = 25
\][/tex]
5. Solve the [tex]\(y\)[/tex]-coordinate Equation:
[tex]\[
7 = \frac{2y + 9}{5}
\][/tex]
Multiply both sides by 5:
[tex]\[
35 = 2y + 9
\][/tex]
Subtract 9 from both sides:
[tex]\[
26 = 2y
\][/tex]
Divide both sides by 2:
[tex]\[
y = 13
\][/tex]
6. Conclusion:
The coordinates of point [tex]\(Q\)[/tex] are [tex]\((25, 13)\)[/tex].
Thus, the answer is:
[tex]\[ \boxed{(25, 13)} \][/tex]
