To solve for the coordinates of point [tex]\(Q\)[/tex] that partitions the directed line segment [tex]\(PR\)[/tex] from [tex]\(P(-10,7)\)[/tex] to [tex]\(R(8,-5)\)[/tex] in the ratio [tex]\(4:5\)[/tex], we use the section formula.
The section formula states that if a point [tex]\(Q\)[/tex] divides a line segment [tex]\(PR\)[/tex] into the ratio [tex]\(m:n\)[/tex], then the coordinates of [tex]\(Q\)[/tex] can be found using:
[tex]\[
Q = \left( \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n} \right)
\][/tex]
Here, the coordinates of the points are:
- [tex]\(P(x_1, y_1) = (-10, 7)\)[/tex]
- [tex]\(R(x_2, y_2) = (8, -5)\)[/tex]
The ratio [tex]\( m:n \)[/tex] is [tex]\( 4:5 \)[/tex]. Thus:
- [tex]\( m = 4 \)[/tex]
- [tex]\( n = 5 \)[/tex]
We apply these values to the section formula.
First, we calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex]:
[tex]\[
Q_x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
= \frac{4 \cdot 8 + 5 \cdot (-10)}{4 + 5}
= \frac{32 - 50}{9}
= \frac{-18}{9}
= -2
\][/tex]
Next, we calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( Q \)[/tex]:
[tex]\[
Q_y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
= \frac{4 \cdot (-5) + 5 \cdot 7}{4 + 5}
= \frac{-20 + 35}{9}
= \frac{15}{9}
= \frac{5}{3}
\][/tex]
Thus, the coordinates of point [tex]\( Q \)[/tex] are:
[tex]\[
Q = \left(-2, \frac{5}{3}\right)
\][/tex]
Therefore, the point [tex]\( Q \)[/tex] that partitions the line segment [tex]\( PR \)[/tex] in the ratio [tex]\( 4:5 \)[/tex] is:
[tex]\[
\boxed{\left(-2, \frac{5}{3}\right)}
\][/tex]
This corresponds to option B.
