Find the point which divides the line segment joining the points (–1, 2) and (4,–5) internally in the ratio 2:3.

Here \$(x_1, y_1) = (2,1), (x_2, y_2) = (4, -5)\$ and \$ m:n = 2:3\$.

The required point is given by

\begin{align} \Bigl( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \Bigr) \\ \end{align}

Substitute the values.

\$\Rightarrow ( \frac{2(4) + 3(-1)}{2 + 3}, \frac{2(-5) + 3(2)}{2 + 3} ) \$

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

\$\Rightarrow ( \frac{8 - 3}{5}, \frac{-10 + 6}{5} ) \$

Add.

\$\Rightarrow ( \frac{5}{5}, \frac{-4}{5} ) \$

Cancel out common factor.

\$\Rightarrow ( \frac{ \cancel{5} ^{1} }{ \cancel{5} ^{1}}, \frac{-4}{5}) \$

Answer

The point which divides the line segment joining the points (–1, 2) and (4,–5) internally in the ratio 2:3 is \$( \frac{1}{1}, \frac{-4}{5} ) \$.