Find the coordinates of the point that divides the join of A(−1, 7) and B(4, −3) in the ratio 2 : 3.

(i) The end points of AB are A(−1, 7) and B(4, −3).
Therefore, (x1 = −1, y1 = 7) and (x2 = 4, y2 = −3)
Also, m = 2 and n = 3
Let the required point be P(x, y).
By section formula, we get:

$x=\frac{\left(m x_{2}+n x_{1}\right)}{(m+n)}, y=\frac{\left(m y_{2}+n y_{1}\right)}{(m+n)}$

$\Rightarrow x=\frac{\{2 \times 4+3 \times(-1)\}}{2+3}, y=\frac{\{2 \times(-3)+3 \times 7\}}{2+3}$

$\Rightarrow x=\frac{8-3}{5}, y=\frac{-6+21}{5}$

$\Rightarrow x=\frac{5}{5}, y=\frac{15}{5}$

Therefore, $x=1$ and $y=3$

Hence, the coordinates of the required point are (1, 3).

(ii) The end points of AB are A(−5, 11) and B(4, −7).
Therefore, (x1 = −5, y1 = 11) and (x2 = 4, y2 = −7).
Also, m = 7 and n = 2
Let the required point be P(x, y).
By section formula, we have:

$x=\frac{\left(m x_{2}+n x_{1}\right)}{(m+n)}, y=\frac{\left(m y_{2}+n y_{1}\right)}{(m+n)}$

$\Rightarrow x=\frac{\{7 \times 4+2 \times(-5)\}}{7+2}, y=\frac{\{7 \times(-7)+2 \times 11\}}{7+2}$

$\Rightarrow x=\frac{28-10}{9}, y=\frac{-49+22}{9}$

$\Rightarrow x=\frac{18}{9}, y=-\frac{27}{9}$

Therefore, $x=2$ and $y=-3$

Hence, the required point is P(2, −3).