Question

The coordinates of the point which divides the line segment joining the points \( (2, -1, 3) \) and \( (4, 3, 1) \) internally in the ratio \( 3:4 \) are:

• A. \( \left( \frac{2}{7}, \frac{20}{7}, \frac{10}{7} \right) \)
• B. \( \left( \frac{10}{7}, \frac{15}{7}, \frac{2}{7} \right) \)
• C. \( \left( \frac{20}{7}, \frac{5}{7}, \frac{15}{7} \right) \)
• D. \( \left( \frac{15}{7}, \frac{20}{7}, \frac{3}{7} \right) \)

Hint:

To find the coordinates of a point dividing a line segment in a given ratio, use the section formula to calculate the weighted average of the coordinates of the endpoints.

Answer 1

The Correct Option is C

Step 1: The formula to find the coordinates of a point dividing a line segment in the ratio \( m : n \) internally is: \[ \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}, \frac{mz_2 + nz_1}{m + n} \right). \] Step 2: Given points \( P(2, -1, 3) \) and \( Q(4, 3, 1) \), and the ratio 3:4, substitute into the formula: \[ x = \frac{3(4) + 4(2)}{3 + 4} = \frac{12 + 8}{7} = \frac{20}{7}, \] \[ y = \frac{3(3) + 4(-1)}{3 + 4} = \frac{9 - 4}{7} = \frac{5}{7}, \] \[ z = \frac{3(1) + 4(3)}{3 + 4} = \frac{3 + 12}{7} = \frac{15}{7}. \] Thus, the coordinates are \( \left( \frac{20}{7}, \frac{5}{7}, \frac{15}{7} \right) \).