Find the coordinates of the point that internally divides the line segment, joining the points (5, 7) and (9, 12) in a ratio 2: 4.

If a point P(x, y) divides the line segment AB joining the points $\mathrm{A}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right) \& \mathrm{B}\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)$ in the ratio m: n then, by the section formula, we have

$\text{x}=\frac{{\text{nx}}_{\text{1}}{\text{+mx}}_{\text{2}}}{\text{m}+\text{n}}$ , $\mathrm{y} = \frac{{\mathrm{ny}}_1 + {\mathrm{my}}_2}{\mathrm{m} + \mathrm{n}}$

Given points: (5, 7) and (9, 12)

Hence $\mathrm{m}=2,\mathrm{n}=4,{\mathrm{x}}_1=5,{\mathrm{x}}_2=9,{\mathrm{y}}_1=7,{\mathrm{y}}_2=12$

$\therefore \mathrm{x}=\frac{(5\times 4)+(9\times 2)}{6} = \frac{20 + 18}{6} = \frac{38}{6}=\frac{19}{3}$

and

$\mathrm{y}=\frac{(7\times 4)+(12\times 2)}{6} = \frac{28 + 24}{6} =\frac{52}{6}= \frac{26}{3}$

Then, the required point $\mathrm{P}(\mathrm{x},\mathrm{y})=\left(\frac{19}{3},\frac{26}{3}\right)$