Find the point dividing the segment joining (−1, 7) and (4, −3) in ratio 2:3.

((2·4 + 3·(−1))/5, (2·(−3) + 3·7)/5) = (1, 3).

Show that (1, 5), (2, 3), (−2, −11) are not collinear.

Compute pairwise distances; sum of two smaller ≠ largest — hence non-collinear.

Take k:1; y-coord = 0 ⇒ (−4k − 6)/(k + 1) = 0 ⇒ k = −3/2 ⇒ externally in 3:2.

Locating points in the plane, distances, section formula, and midpoint results.

Cartesian plane: x-axis, y-axis, origin O(0,0). Any point represented by ordered pair (x, y).
Quadrants: I (+,+), II (−,+), III (−,−), IV (+,−).
Distance formula between P(x₁, y₁) and Q(x₂, y₂): PQ = √[(x₂ − x₁)² + (y₂ − y₁)²].
Distance from origin O to P(x, y) = √(x² + y²).
Section formula (internal division): point dividing segment joining (x₁, y₁) and (x₂, y₂) in ratio m:n is ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)).
Midpoint formula (m = n = 1): ((x₁ + x₂)/2, (y₁ + y₂)/2).
Three points are collinear if the area of the triangle formed is 0 — but the direct area formula for triangles was rationalised out; use section-based reasoning or slope idea inferred from similar triangles.
For a parallelogram, diagonals bisect each other → equate midpoints of the two diagonals.
Common problem types: find distance, find ratio in which a point divides a segment, find fourth vertex of a parallelogram, verify collinearity, find coordinates of a specific point (like centroid — using midpoint reasoning).
For ratio problems, take unknown ratio k:1, apply section formula, solve for k.

Q. Find the point dividing the segment joining (−1, 7) and (4, −3) in ratio 2:3.
A. ((2·4 + 3·(−1))/5, (2·(−3) + 3·7)/5) = (1, 3).
Q. Show that (1, 5), (2, 3), (−2, −11) are not collinear.
A. Compute pairwise distances; sum of two smaller ≠ largest — hence non-collinear.
Q. Find the ratio in which x-axis divides the join of (5, −6) and (−1, −4).
A. Take k:1; y-coord = 0 ⇒ (−4k − 6)/(k + 1) = 0 ⇒ k = −3/2 ⇒ externally in 3:2.