Respuesta :
The coordinates of point Z are (–7, 1).
Solution:
Given data: X(–2, 6) and Y(–10, –2)
Point Z partitions the line segment XY in the ratio 5:3.
XZ : ZY = 5 : 3
X(–2, 6) can be taken as [tex]X(x_1, y_1)[/tex].
Y(–10, –2) can be taken as [tex]Y(x_2, y_2)[/tex].
XZ : ZY can be taken as m : n = 5 : 3.
We know that coordinate of point [tex]Z(x_3, y_3)[/tex] divides line segment joining [tex]A(x_1, y_1)[/tex] and [tex]B(x_2, y_2)[/tex] in ratio m : n is
[tex]Z(x_3,y_3)=(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n})[/tex]
Here, [tex]x_1 = -2, x_2=-10, y_1=6, y_2=-2[/tex] and m = 5, n = 3.
Substitute these in the above formula, we get,
[tex]Z\left(x_{3}, y_{3}\right)=\left(\frac{5 \times(-10)+3 \times(-2)}{5+3}, \frac{5 \times(-2)+3 \times(6)}{5+3}\right)[/tex]
⇒ [tex]=\left(\frac{-50-6}{8}, \frac{-10+18}{8}\right)[/tex]
⇒ [tex]=\left(\frac{-56}{8}, \frac{8}{8}\right)[/tex]
⇒ [tex]Z\left(x_{3}, y_{3}\right)=(-7,1)[/tex]
Hence the coordinates of point Z are (–7, 1).
