If A and B are (−2, −2) and (2, −4), respectively, find the coordinates of P such that \(AP = \dfrac{3}{7} AB\) and P lies on the line segment AB.
P = \(\left(-\dfrac{2}{7}, -\dfrac{20}{7}\right)\)
Let the coordinates of A and B be \(A(-2, -2)\) and \(B(2, -4)\). We are told that point P lies on segment AB and that \(AP = \tfrac{3}{7} AB\).
This means P divides AB internally in the ratio \(AP : PB = 3 : 4\), because the whole length AB is split into 3 parts from A to P and 4 parts from P to B.
Using the section formula, if a point P divides the line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) internally in the ratio \(m : n\), then its coordinates are \(\left(\dfrac{nx_1 + mx_2}{m + n}, \dfrac{ny_1 + my_2}{m + n}\right)\).
Here, \(x_1 = -2, y_1 = -2, x_2 = 2, y_2 = -4\) and \(m = 3, n = 4\). So
\(x_P = \dfrac{4(-2) + 3(2)}{3 + 4} = \dfrac{-8 + 6}{7} = \dfrac{-2}{7}\).
\(y_P = \dfrac{4(-2) + 3(-4)}{3 + 4} = \dfrac{-8 - 12}{7} = \dfrac{-20}{7}\).
Therefore, the coordinates of P are \(\left(-\dfrac{2}{7}, -\dfrac{20}{7}\right)\).