Locus
A lightweight summary of everything you need to know about locus conditions, formulas, and common exam traps for DSE Mathematics (Compulsory Part).
1. The Core Concept
What is a Locus? A locus is simply a path traced by a moving point P(x, y) that must follow a specific rule (condition). In the DSE, this path always turns into either a straight line (
Ax + By + C = 0) or a circle (x² + y² + Dx + Ey + F = 0).2. DSE Locus Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Equidistant from 2 points A and B | Perpendicular bisector of AB | Midpoint of AB + negative reciprocal slope:mbisector = −1 / mAB |
| 2. Fixed distance r from a point C(h, k) | Circle | Standard Form:(x − h)² + (y − k)² = r² |
| 3. Fixed distance d from a straight line L | A pair of parallel lines on both sides | Same slope as L, distance calculated via point-to-line formula. |
| 4. Equidistant from 2 intersecting lines | A pair of angle bisectors | Point-to-line distance equation:|A₁x + B₁y + C₁| / √(A₁² + B₁²) = |A₂x + B₂y + C₂| / √(A₂² + B₂²) |
| 5. Combined Conditions (Section B) | Intersection points of a circle & a line | Substitute y = mx + c into the circle equation and solve using the discriminant (Δ). |
Click the graph to explore it interactively →
💡 Exam Traps & Secrets
⚠️ The "Pair of Lines" TrapFor Case 3 (fixed distance from a line) and Case 4 (angle bisectors), the locus is always a pair of lines. If you only write one equation in your Paper 1 exam, you will lose half the marks.
⚠️ The "Does Not Exist" TrapSometimes Section B questions ask for the intersection of two loci. If the straight line does not touch the circle (i.e., Δ < 0), the locus of the combined condition does not exist. You must explicitly state this to secure the final mark.
✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Perpendicular Bisector)Find the locus of P(x, y) equidistant from A(−2, 3) and B(4, 1).
(Circle Equation)Find the locus of P(x, y) always at distance 5 from the origin (0, 0).
(Angle Bisector)Find the locus of P equidistant from
3x − 4y + 1 = 0 and x + 2y − 5 = 0.