The line segment joining the points $A(4, -5)$ and $B(4, 5)$ is divided by the point $P$ such that $AP : AB = 2 : 5$. Find the coordinates of $P$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:30 · grounding rag
Model Answer
Given: A(4, −5), B(4, 5), and AP : AB = 2 : 5.
Since AP : AB = 2 : 5, we get AP : PB = 2 : 3.
Using the section formula with $m_1 : m_2 = 2 : 3$:
$$x = \frac{2(4) + 3(4)}{2+3} = \frac{8+12}{5} = 4$$
$$y = \frac{2(5) + 3(-5)}{2+3} = \frac{10-15}{5} = -1$$
∴ Coordinates of P are (4, −1).
Source: Chapter 7, Section 7.3 — Section Formula
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Explanation
- The key step is converting AP : AB = 2 : 5 into AP : PB = 2 : 3 (since PB = AB − AP = 5 − 2 = 3 parts).
- Then apply the section formula directly. Examiners award 1 mark for finding the correct ratio (2 : 3) and 1 mark for the correct final coordinates.
- Since both x-coordinates of A and B are 4, the x-coordinate of P must also be 4 — a useful check.