Determine the coordinates of the point that divides the line segment connecting $(5,-2)$ and $(9,6)$ in the ratio $3:1.$

If a point (x, y) divides the line joining the points $\left({\text{x}}_1,{\text{y}}_1\right)$and $\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)$ in the ratio m: n, then by section formula:

$(\mathrm{x},\mathrm{y})=\left(\frac{{\mathrm{mx}}_2+{\mathrm{nx}}_1}{\mathrm{m}+\mathrm{n}},\frac{{\mathrm{my}}_2+{\mathrm{ny}}_1}{\mathrm{m}+\mathrm{n}}\right)$

Let the points be A(5, - 2) and B(9, 6). Let a point P(x, y) divide AB in the ratio 3: 1.

Here, m = 3, n = 1, x₁ = 5, y₁ = 9, x₂ = -2 and y₂ = 6.

On substitution, we get:

$\mathrm{P}(\mathrm{x},\mathrm{y})=\left(\frac{3\times 9+1\times 5}{3+1},\frac{3\times 6+1\times (-2)}{3+1}\right)$

$\mathrm{P}(\mathrm{x},\mathrm{y})=\left(\frac{27+5}{4},\frac{18-2}{4}\right)$

$\mathrm{P}(\mathrm{x},\mathrm{y})=\left(\frac{32}{4},\frac{16}{4}\right)$

$\mathrm{P}(\mathrm{x},\mathrm{y})=\left(8,4\right)$

Hence, the required point is (8,4).