Question

Find the coordinates of the point which divides the line segment joining the points (4, -3) and (8, 5) in the ratio 3 : 1 internally.

Hint:

Always substitute carefully in the section formula: \( P(x, y) = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \).

Answer 1

Step 1: Recall the section formula.
If a point \( P(x, y) \) divides the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) internally in the ratio \( m : n \), then \[ P(x, y) = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \]
Step 2: Substitute the given values.
\[ x_1 = 4, \; y_1 = -3, \; x_2 = 8, \; y_2 = 5, \; m = 3, \; n = 1 \]
Step 3: Find the coordinates.
\[ x = \frac{(3)(8) + (1)(4)}{3 + 1} = \frac{24 + 4}{4} = 7 \] \[ y = \frac{(3)(5) + (1)(-3)}{3 + 1} = \frac{15 - 3}{4} = 3 \]
Step 4: Conclusion.
Hence, the coordinates of the required point are \( \boxed{(7, 3)} \).