Circle
A conic section defined as the set of points equidistant from a centre.
Standard Equation
Centre $(h,k)$, radius $r$: $$ (x - h)^2 + (y - k)^2 = r^2 $$
General Equations
Standard general form: $$ x^2 + y^2 + Ax + By + C = 0 $$ where centre is $\left(-\frac{A}{2}, -\frac{B}{2}\right)$ and radius $r = \sqrt{\frac{A^2}{4} + \frac{B^2}{4} - C}$.
Lecture form (coefficients tied to centre coordinates): $$ x^2 - 2hx + y^2 - 2ky + C = 0 \quad ; \quad C = h^2 + k^2 - r^2 $$ where $(h,k)$ is the centre and: $$ r = \sqrt{h^2 + k^2 - C} $$
Intersection with a Straight Line
Solve the line and circle equations simultaneously. The discriminant determines the geometric relationship:
| Discriminant | Roots | Relationship |
|---|---|---|
| $\Delta > 0$ | Two distinct real roots | Line cuts circle at two points |
| $\Delta = 0$ | One repeated real root | Line is tangent to the circle |
| $\Delta < 0$ | No real roots | Line does not intersect the circle |
Tangent and Normal
Tangent at a point: perpendicular to the radius at the point of contact.
Normal at a point: perpendicular to the tangent (passes through the centre).
Length of tangent from external point $(m,n)$ to circle with centre $(h,k)$ and radius $r$: $$ ST = \sqrt{(m-h)^2 + (n-k)^2 - r^2} $$
Equation from diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$: $$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$$
Conic Section Relationships
flowchart LR
CONICS((Conic Sections))
CONICS --> ELLIPSE[Ellipse<br/>Sum of distances to foci is constant]
CONICS --> PARABOLA[Parabola<br/>Equidistant from focus and directrix]
CONICS --> HYPERBOLA[Hyperbola<br/>Difference of distances to foci is constant]
ELLIPSE --> CIRCLE[Circle<br/>Special case: a = b]
Identifying Conic Sections
flowchart TD
A["General Equation: Ax² + Cy² + Dx + Ey + F = 0"] --> B{"Are both A and C present?"}
B -->|"Yes, same sign"| C[Ellipse family]
B -->|"Yes, opposite signs"| D[Hyperbola]
B -->|"Only one present"| E[Parabola]
C --> F{"A = C?"}
F -->|"Yes"| G[Circle]
F -->|"No"| H[Ellipse]