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]

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