Position of two points relative to a line — A/L Combined Mathematics

What must I remember about Position of two points relative to a line?

• Linear expression L(x, y) = ax + by + c defines three regions in the xy-plane: the straight line itself (L(x, y) = 0), and two open half-planes (L(x, y) > 0 and L(x, y) < 0).
• Relative position test: Two points P(x1, y1) and Q(x2, y2) lie on the SAME side of line L: ax + by + c = 0 if and only if L(x1, y1) and L(x2, y2) have the SAME sign: (ax1 + by1 + c)(ax2 + by2 + c) > 0.
• Opposite sides test: Points P and Q lie on OPPOSITE sides of line L if and only if L(x1, y1) and L(x2, y2) have OPPOSITE signs: (ax1 + by1 + c)(ax2 + by2 + c) < 0.
• Point on line: Point P lies strictly ON the line if and only if L(x1, y1) = 0.
• Position relative to the origin (0, 0): Point P lies on the same side as origin if L(x1, y1) and c have the same sign (c * L(x1, y1) > 0), and on opposite side of origin if c * L(x1, y1) < 0.

Test yourself (1 minute)

1. Ratio of division of line segment by line

   The line L: ax + by + c = 0 divides the line segment joining P(x1, y1) and Q(x2, y2) in the ratio m : n = - L(x1, y1) / L(x2, y2); internal division if ratio > 0 (opposite signs), external division if ratio < 0 (same signs).

2. Point inside a triangle

   Point P(x0, y0) lies strictly INSIDE triangle ABC if P and vertex A lie on the same side of line BC, P and B lie on same side of line AC, and P and C lie on same side of line AB.

3. Linear programming feasible regions

   Systems of linear inequalities like ax + by + c ≥ 0 define convex polygonal feasible regions bounded by half-planes.

Practise Position of two points relative to a line questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 18

What should I study first?

What comes next?

Next in the syllabus

Where does Position of two points relative to a line fit in the syllabus?

Year 1 · Term 3