To determine the ratio in which the point [tex]\( P(2, -5) \)[/tex] divides the line segment joining [tex]\( A(-3, 5) \)[/tex] and [tex]\( B(4, -9) \)[/tex], we use the section formula. The section formula states that if a point [tex]\( P(x, y) \)[/tex] divides the line segment joining points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in a ratio [tex]\( m:n \)[/tex], then:
[tex]\[
P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Given:
- [tex]\( P = (2, -5) \)[/tex]
- [tex]\( A = (-3, 5) \)[/tex]
- [tex]\( B = (4, -9) \)[/tex]
We need to find the ratio [tex]\( m:n \)[/tex].
First, let's setup the equations using the section formula:
1. For the x-coordinate:
[tex]\[
\frac{mx_2 + nx_1}{m + n} = x_P
\][/tex]
Substituting the given points:
[tex]\[
\frac{m \cdot 4 + n \cdot (-3)}{m + n} = 2
\][/tex]
This leads to:
[tex]\[
4m - 3n = 2(m + n)
\][/tex]
Simplifying this:
[tex]\[
4m - 3n = 2m + 2n
\][/tex]
[tex]\[
4m - 2m = 3n + 2n
\][/tex]
[tex]\[
2m = 5n
\][/tex]
[tex]\[
\frac{m}{n} = \frac{5}{2}
\][/tex]
2. For the y-coordinate:
[tex]\[
\frac{my_2 + ny_1}{m + n} = y_P
\][/tex]
Substituting the given points:
[tex]\[
\frac{m \cdot (-9) + n \cdot 5}{m + n} = -5
\][/tex]
This leads to:
[tex]\[
-9m + 5n = -5(m + n)
\][/tex]
Simplifying this:
[tex]\[
-9m + 5n = -5m - 5n
\][/tex]
[tex]\[
-9m + 5m = -5n - 5n
\][/tex]
[tex]\[
-4m = -10n
\][/tex]
[tex]\[
4m = 10n
\][/tex]
[tex]\[
\frac{m}{n} = \frac{10}{4} = \frac{5}{2}
\][/tex]
Therefore, both x and y coordinates confirm that the ratio [tex]\( m:n \)[/tex] is [tex]\( \frac{5}{2} \)[/tex]. Thus, the point [tex]\( P(2, -5) \)[/tex] divides the line segment joining [tex]\( A(-3, 5) \)[/tex] and [tex]\( B(4, -9) \)[/tex] in the ratio [tex]\( 5:2 \)[/tex].
