Find the point which divides the line segment joining the points (2,1) and (3,5) externally in the ratio 2:3.

Here \$(x_1, y_1) = (2,1), (x_2, y_2) = (3,5)\$, and \$m:n = 2:3 ( m < n)\$.

The required point is given by

\$( \frac{nx_1 - mx_2}{n - m}, \frac{ny_2 - my_2}{n - m} ) \$

Substiutute the values.

\$\Rightarrow ( \frac{3(2) - 2(3)}{3 - 2}, \frac{3(1) - 2(5)}{3 - 2}) \$

Multiply in the numerator and sum the values in the denominator.

\$ \Rightarrow ( \frac{6 - 6}{1}, \frac{3 - 10}{1} ) \$

Subtract.

\$\Rightarrow ( \frac{0}{1}, \frac{-7}{1} ) \$

Answer

The point which divides the line segment joining the points (2,1) and (3,5) internally in the ratio 2:3 is \$( 0, -7). \$