To find the coordinates of the point \( Q \) that divides the line segment \( PR \) into the given ratio, we can use the section formula. The section formula for internal division of a line segment in a given ratio can be expressed as:
[tex]\[
Q \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
where \((x_1, y_1)\) are the coordinates of point \( P \), \((x_2, y_2)\) are the coordinates of point \( R \), \(m\) is the first part of the ratio, and \(n\) is the second part.
Given:
[tex]\[
P(-10, 7)
\][/tex]
[tex]\[
R(8, -5)
\][/tex]
and the ratio is 4.5 to 1, which means \( m = 4.5 \) and \( n = 1 \).
First, we will find the x-coordinate \( Q_x \) of point \( Q \):
[tex]\[
Q_x = \frac{(m \cdot x_2) + (n \cdot x_1)}{m+n}
\][/tex]
Substituting the values, we get:
[tex]\[
Q_x = \frac{(4.5 \cdot 8) + (1 \cdot -10)}{4.5 + 1}
\][/tex]
[tex]\[
Q_x = \frac{36 - 10}{5.5}
\][/tex]
[tex]\[
Q_x = \frac{26}{5.5}
\][/tex]
[tex]\[
Q_x \approx 4.7272727272727275
\][/tex]
Next, we will find the y-coordinate \( Q_y \) of point \( Q \):
[tex]\[
Q_y = \frac{(m \cdot y_2) + (n \cdot y_1)}{m+n}
\][/tex]
Substituting the values, we get:
[tex]\[
Q_y = \frac{(4.5 \cdot -5) + (1 \cdot 7)}{4.5 + 1}
\][/tex]
[tex]\[
Q_y = \frac{-22.5 + 7}{5.5}
\][/tex]
[tex]\[
Q_y = \frac{-15.5}{5.5}
\][/tex]
[tex]\[
Q_y \approx -2.8181818181818183
\][/tex]
Therefore, the coordinates of point \( Q \) are approximately \( (4.7272727272727275, -2.8181818181818183) \).
Among the provided options, none match the coordinates obtained. Therefore, there could be a mistake in the options given, but the coordinates of [tex]\( Q \)[/tex] based on the calculations are correct and are approximately [tex]\( (4.7272727272727275, -2.8181818181818183) \)[/tex].
