(8.4, 8.4) or [tex]\left(\frac{42}{5}, \frac{42}{5}\right)[/tex]
Explanation:
Given points are A(2, 6) and B(18, 12).
Let P(x, y) partitions the line segment in the ratio 2 : 3.
That is AP : PB = 2 : 3.
A(2, 6) can be taken as [tex]A(x_1, y_1).[/tex]
B(18, 12) can be taken as [tex]B(x_2, y_2).[/tex]
AP : PB can be taken as m : n = 2 : 3.
The coordinate of point P(x, y) 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]P(x,y)=(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n})[/tex]
Here, [tex]x_1 = 2, x_2=18, y_1=6, y_2=12[/tex] and m = 2, n = 3.
Substitute these in the above formula, we get
[tex]P\left(x, y\right)=\left(\frac{2 \times 18+3 \times 2}{2+3}, \frac{2 \times12+3 \times 6}{2+3}\right)[/tex]
[tex]=\left(\frac{36+6}{5}, \frac{24+18}{5}\right)[/tex]
[tex]=\left(\frac{42}{5}, \frac{42}{5}\right)[/tex]
= (8.4, 8.4)
Hence, the point partitions the line segment is (8.4, 8.4) or [tex]\left(\frac{42}{5}, \frac{42}{5}\right)[/tex].
