Equations of Circles
Master standard and general circle forms, find centers and radii, and learn the fastest methods to solve circle-line intersections on the DSE.
1. The Core Concept
What is a Circle Equation? A circle is defined as the set of all points (x, y) that are at a constant distance (the radius, r) from a fixed point (the center,
(h, k)). In the DSE, you must be comfortable switching between Standard Form (which tells you the center and radius immediately) and General Form (which requires algebraic formulas or completing the square).2. DSE Equations of Circles 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. Standard Form | (x − h)² + (y − k)² = r²• Centre: (h, k)• Radius: r | Tip: Pay attention to the signs.(x + 3)² ⟹ h = −3. |
| 2. General Form | x² + y² + Dx + Ey + F = 0• Centre: (−D/2, −E/2)• Radius: r = √(D²/4 + E²/4 − F) | Real circle condition:D² + E² − 4F > 0. If ≤ 0, no real circle exists. |
| 3. Diameter Form | Given endpoints A(x₁, y₁) and B(x₂, y₂):(x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0 | A massive time-saver for Paper 2. You do not need to find the midpoint or radius to write this equation. |
| 4. Point vs. Circle | Compare distance d from point P to centre C with radius r: • d > r: Outside the circle • d = r: On the circle • d < r: Inside the circle | Alternatively, substitute P(x₁, y₁) into the general equation:• Result > 0: Outside • Result = 0: On • Result < 0: Inside |
| 5. Line Intersections | Substitute y = mx + c into the circle equation. Solve the resulting quadratic using the discriminant (Δ = b² − 4ac).• Δ > 0: 2 intersection points • Δ = 0: 1 intersection point (Tangent) • Δ < 0: 0 intersection points | For tangents, use the geometric shortcut: perpendicular distance from centre to line = radius (d = r). Much faster than solving Δ = 0. |
.webp)
Click the graph to explore it interactively →
💡 Exam Traps & Secrets
⚠️ The Coefficient Check (x² and y²)You cannot read D, E, and F directly from a general equation if the coefficients of x² and y² are not 1. Example: If given
3x² + 3y² − 12x + 18y − 9 = 0, you must divide the entire equation by 3 first to get x² + y² − 4x + 6y − 3 = 0. Only then can you state D = −4, E = 6, F = −3.⚠️ The Imaginary Circle Condition (Paper 2)For any general equation to represent a real circle, the radius calculation must yield a real number. This means the value inside the square root must be strictly positive:
D² + E² − 4F > 0. If D² + E² − 4F = 0, the equation represents a single point. If it is < 0, no real circle exists.⚠️ The Geometric Tangent ShortcutWhen finding the tangent to a circle, substituting
y = mx + c and solving Δ = 0 is very slow. Shortcut: Calculate the perpendicular distance d from the circle's center to the line using the point-to-line formula. Set d = r (distance equals radius). This is usually twice as fast as the algebraic Δ method.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(General to Standard Form conversion)Find the centre and radius of the circle
x² + y² − 6x + 8y − 11 = 0.(Quadratic Line-Circle Intersection)Find the points of intersection of the straight line
y = 2x + 1 and the circle x² + y² = 25.(Position of a Point)Determine if the point P
(5, −2) lies inside, on, or outside the circle (x − 2)² + (y + 1)² = 16.