In what ratio does the point (−4, 6) internally divide the line segment…
2022
In what ratio does the point (−4, 6) internally divide the line segment joining A (−6, 10) and B (3, −8)?
Answer: A. 2 : 7 — ConceptIf a point P divides the segment from A(x₁, y₁) to B(x₂, y₂) internally in the ratio AP:PB = m:n, the section formula forms a weighted average of the…
- A.
2 : 7
- B.
3 : 7
- C.
4 : 7
- D.
5 : 7
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Correct answer: A
Concept
If a point P divides the segment from A(x₁, y₁) to B(x₂, y₂) internally in the ratio AP:PB = m:n, the section formula forms a weighted average of the endpoints.
Its coordinates are x = (m x₂ + n x₁)/(m+n) and y = (m y₂ + n y₁)/(m+n). The weights are opposite the endpoints they multiply.
Application
Let AP:PB = m:n. Substitute A(−6, 10), B(3, −8), and P(−4, 6) into the x-coordinate formula: −4 = [3m − 6n]/(m+n).
Multiply by m+n: −4m − 4n = 3m − 6n.
Collect like terms: 2n = 7m, so m/n = 2/7 and therefore m:n = 2:7.
Thus the point divides AB internally in the ratio AP:PB = 2:7.
Cross-check
Using m:n = 2:7 in the y-coordinate formula gives y = [2(−8) + 7(10)]/9 = 54/9 = 6, exactly the given y-coordinate. This independently confirms the ratio.
Result
Therefore, the required ratio is 2:7.