Section Formula Applications
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Direct answer
Divide the segment from P(x1, y1) to Q(x2, y2) in the ratio m : n and the point is ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)) internally — externally, the plus in the denominator turns to minus. The surrounding family does the real work: midpoints are the m = n case, the centroid is ((x1 + x2 + x3)/3, ...), the incentre weights vertices by opposite side lengths, and each excentre carries exactly one negative weight. The formula also runs in reverse: when a line cuts a given segment, assume the unknown ratio λ : 1, write the dividing point, and force it onto the line — one substitution, one linear equation, one ratio, sign included.
What you must remember
- Internal division: ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)) for ratio m : n measured from the first point.
- External division: ((mx2 − nx1)/(m − n), (my2 − ny1)/(m − n)) — the point lies beyond one end, and the formula delivers it without separate reasoning.
- Centroid: the average of the three vertices; it trisects each median, dividing it 2 : 1 from vertex to side.
- Incentre: weights (a, b, c) — the side lengths opposite the vertices — never the coordinates of anything; the excentre opposite A uses weights (−a, b, c).
- Ratio-finding template: a point on segment PQ at ratio λ : 1 has coordinates ((λx2 + x1)/(λ + 1), (λy2 + y1)/(λ + 1)); substitute into any line or curve condition and solve for λ.
- Sign of λ: negative λ means the point divides externally, and the magnitude gives the ratio — the interpretation step after the algebra.
- Harmonic partner: the internal and external division points of the same ratio are harmonic conjugates with respect to the segment's ends — vocabulary Advanced papers use.
A line dividing a segment
Find the ratio in which the line 2x + 3y − 5 = 0 divides the segment joining A(1, 2) and B(4, 6). Assume the ratio λ : 1 measured from A, so the dividing point is ((4λ + 1)/(λ + 1), (6λ + 2)/(λ + 1)). Substituting into the line: 2(4λ + 1) + 3(6λ + 2) = 5(λ + 1), which expands to 26λ + 8 = 5λ + 5, so 21λ = −3 and λ = −1/7. The negative sign reads as external division in the ratio 1 : 7 — the line meets the segment extended, not the segment itself. The point confirms it: substituting λ = −1/7 gives (1/2, 4/3), and 2(1/2) + 3(4/3) = 1 + 4 = 5 sits on the line exactly, while the point lies beyond A away from B. This template — assume, substitute, solve, interpret the sign — is the entire technique, and it transfers unchanged to circles, conics and any locus asked to divide a given segment.
Signs and ratios
JEE Main asks the direct divisions — centroid, midpoint, a stated ratio — and the ratio-finding template above, where the standard distractor reports only |λ| = 1/7 and calls it internal. The m : n order is the other chronic leak: the ratio 2 : 3 from A means m multiplies the B-end coordinates, and reversing the assignment moves the point to the mirror position inside the segment. JEE Advanced prefers the weighted centres: incentre and excentre coordinates with side-length weights, often combined with a triangle whose sides must first be computed from the vertices by the distance formula — two chapters chained in one item. The centroid-versus-incentre confusion is deliberately planted: the centroid averages coordinates, the incentre averages them weighted by opposite sides, and only in the equilateral case do the two coincide. A final Advanced flourish divides a segment in a ratio involving a parameter and asks for the parameter making the division point lie on a given curve — the same λ-substitution with a quadratic at the end instead of a linear one.
Frequently asked questions
What is the section formula for internal division?
((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)) for ratio m : n, with m weighting the second point's coordinates.
How does external division differ?
Denominator m − n and signs flipped in the numerator: ((mx2 − nx1)/(m − n), (my2 − ny1)/(m − n)).
What are the incentre's coordinates?
((ax1 + bx2 + cx3)/(a + b + c), (ay1 + by2 + cy3)/(a + b + c)) with a, b, c the side lengths opposite the respective vertices.
How do you find the ratio in which a line cuts a segment?
Assume ratio λ : 1, write the dividing point, substitute into the line's equation and solve — a negative λ signals external division.
Where is the centroid and how does it divide medians?
At the coordinate average ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3), trisecting each median in the ratio 2 : 1 from the vertex.