Question

Find the coordinates of the point which divides the line segment joining A(3, –2) and B(5, 4) in the ratio 2:3.

Hint:

Use the section formula carefully and substitute values for \(x_1, y_1, x_2, y_2\) and the ratio precisely.

Answer 1

Step 1: Use the section formula for internal division: \[ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \] Step 2: Assign values: \[ x_1 = 3, \ y_1 = -2, \quad x_2 = 5, \ y_2 = 4, \quad m:n = 2:3 \] Step 3: Calculate coordinates: \[ x = \frac{2 \cdot 5 + 3 \cdot 3}{2 + 3} = \frac{10 + 9}{5} = \frac{19}{5}
y = \frac{2 \cdot 4 + 3 \cdot (-2)}{5} = \frac{8 - 6}{5} = \frac{2}{5} \] Final Answer: \[ \boxed{\left( \frac{19}{5}, \frac{2}{5} \right)} \]