Find the ratio in which the point $P(-4, 6)$ divides the line segment joining the points $A(-6, 10)$ and $B(3, -8)$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:29 · grounding rag
Model Answer
Let P(–4, 6) divide AB internally in the ratio $k : 1$.
By the section formula:
$$-4 = \frac{3k + (-6)}{k + 1}$$
$$-4k - 4 = 3k - 6$$
$$7k = 2 \implies k = \frac{2}{7}$$
So the ratio is $k : 1 = \mathbf{2 : 7}$.
Verification with y-coordinate:
$$y = \frac{-8(2) + 10(7)}{2 + 7} = \frac{-16 + 70}{9} = \frac{54}{9} = 6 \checkmark$$
Therefore, P(–4, 6) divides AB in the ratio 2 : 7.
Source: Chapter 7, Section 7.3 — Section Formula (Example 7)
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Explanation
- Use the section formula with ratio $k:1$ (one unknown makes algebra faster).
- Apply it to the x-coordinate to find $k$, then verify with the y-coordinate — examiners award marks for verification.
- State the final ratio clearly. Writing both the working and the check is expected for full 2 marks.