In what ratio does the point (– 4, 6) divide the line segment joining the points A(– 6, 10) and B(3, – 8)?
Step-by-Step Solution
Key Concept: Use the section formula (internal division) for coordinates of a point dividing a line segment in a given ratio. If a point P(x, y) divides AB joining A(x₁, y₁) and B(x₂, y₂) in the ratio m:n, then \(x = \frac{mx_2 + nx_1}{m+n}\) and \(y = \frac{my_2 + ny_1}{m+n}\). Solve these two equations for m and n.
1. Assign symbols\
Let the required ratio be \(AP : PB = m : n\) (internal division).\
Coordinates: \(A(x_1, y_1) = (-6, 10)\), \(B(x_2, y_2) = (3, -8)\), \(P(x, y) = (-4, 6)\).\
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2. Write the section‑formula equations\
\[\begin{aligned}
x &= \frac{m x_2 + n x_1}{m+n} = \frac{3m + (-6)n}{m+n},\\[4pt]
y &= \frac{m y_2 + n y_1}{m+n} = \frac{-8m + 10n}{m+n}.
\end{aligned}\]\
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3. Substitute the known coordinates\
\[\begin{aligned}
-4 &= \frac{3m - 6n}{m+n} \quad\Rightarrow\quad -4(m+n) = 3m - 6n,\\[4pt]
6 &= \frac{-8m + 10n}{m+n} \quad\Rightarrow\quad 6(m+n) = -8m + 10n.
\end{aligned}\]\
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4. Simplify each equation\
- From the first equation:\
\[-4m - 4n = 3m - 6n \Rightarrow -7m + 2n = 0 \Rightarrow 2n = 7m \Rightarrow n = \frac{7}{2}m.\
- From the second equation:\
\[6m + 6n = -8m + 10n \Rightarrow 14m - 4n = 0 \Rightarrow 4n = 14m \Rightarrow n = \frac{7}{2}m.\
Both equations give the same relation, confirming consistency.
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5. Find the ratio\
Since \(n = \frac{7}{2}m\),\
\[\frac{m}{n} = \frac{m}{\frac{7}{2}m} = \frac{2}{7}.\]
Hence \(m:n = 2:7\).
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6. Conclusion\
The point \((-4,6)\) divides the segment \(AB\) internally in the ratio \(AP : PB = 2 : 7\).
Correct Answer: 2 : 7