Question

Find the coordinates of the point which divides the line segment joining the points A(-1, 7) and B(4, -3) in the ratio 2 : 3.

Hint:

Use the section formula $\left( \dfrac{mx_2 + nx_1}{m + n}, \dfrac{my_2 + ny_1}{m + n} \right)$ for dividing a line segment internally.

Answer 1

Step 1: Given.
$A(x_1, y_1) = (-1, 7)$, \; $B(x_2, y_2) = (4, -3)$, \; ratio $m : n = 2 : 3$.
Step 2: Use the section formula.
For internal division, the coordinates $(x, y)$ are: \[ x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n} \] Step 3: Substitute values.
\[ x = \frac{(2)(4) + (3)(-1)}{2 + 3} = \frac{8 - 3}{5} = \frac{5}{5} = 1 \] \[ y = \frac{(2)(-3) + (3)(7)}{2 + 3} = \frac{-6 + 21}{5} = \frac{15}{5} = 3 \]
Step 4: Conclusion.
The required point is $(1, 3)$.