Parabola

A conic section defined as the set of points equidistant from a focus and a directrix.

Derivation

When vertex is at origin, focus $F(0,a)$, directrix $y = -a$: $$\sqrt{x^2 + (y-a)^2} = y + a$$ $$x^2 + (y-a)^2 = (y+a)^2$$ $$x^2 = 4ay$$

When vertex is at $(h,k)$, focus $F(h, k+a)$, directrix $y = k-a$: $$\sqrt{(x-h)^2 + (y-(k+a))^2} = |y-(k-a)|$$ $$(x-h)^2 = 4a(y-k)$$

Important Terms

  • Focus: fixed point, $a$ units from the vertex on the axis of the parabola
  • Directrix: fixed line perpendicular to the axis, $a$ units from the vertex
  • Axis: line through the focus and vertex, also perpendicular to the directrix
  • Vertex: point of intersection between parabola and axis; midpoint of focus and directrix
  • Latus rectum: chord through the focus parallel to the directrix

For $x^2 = 4ay$, the points on the parabola with $y = a$ are $(\pm 2a, a)$. Length of latus rectum: $$\sqrt{(2a - (-2a))^2 + (a-a)^2} = 4a$$

Standard Equations

Orientation Equation Vertex Focus Directrix Shape
Vertical $(x-h)^2 = 4a(y-k); a>0$ $(h,k)$ $(h, k+a)$ $y = k-a$ Opens upward
Vertical $(x-h)^2 = 4a(y-k); a<0$ $(h,k)$ $(h, k-a)$ $y = k+a$ Opens downward
Horizontal $(y-k)^2 = 4a(x-h); a>0$ $(h,k)$ $(h+a, k)$ $x = h-a$ Opens to the right
Horizontal $(y-k)^2 = 4a(x-h); a<0$ $(h,k)$ $(h-a, k)$ $x = h+a$ Opens to the left

General Form

The general form can be transformed to standard form by completing the square.

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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