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.
Step-by-Step Solution
Key Concept: Use the section formula for internal division of a line segment.
1. Let the required point be P(x, y) which divides the segment joining A(4, -3) and B(8, 5) in the ratio 3:1 internally.
2. For internal division, the section formula is
$$x = \frac{m x_2 + n x_1}{m+n}, \quad y = \frac{m y_2 + n y_1}{m+n}$$
where m:n = 3:1, (x_1, y_1) = (4, -3) and (x_2, y_2) = (8, 5).
3. Substitute the given values:
$$x = \frac{3\times8 + 1\times4}{3+1} = \frac{24+4}{4} = 7$$
$$y = \frac{3\times5 + 1\times(-3)}{3+1} = \frac{15-3}{4} = 3$$
4. Therefore, the coordinates of the required point are (7, 3).
Correct Answer: (7, 3)