Ellipse

A conic section defined as the set of all points $P(x,y)$ in a plane such that the sum of distances to two fixed points (the foci) is constant ($2a$).

Derivation

Place the centre at the origin with foci at $(-c,0)$ and $(c,0)$. For any point $P(x,y)$ on the ellipse:

$$d_1 + d_2 = 2a$$ $$\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a$$

After isolating one radical, squaring twice, and simplifying:

$$\left(1 - \frac{c^2}{a^2}\right)x^2 + y^2 = a^2 - c^2$$

Dividing by $a^2 - c^2$:

$$\frac{x^2}{a^2} + \frac{y^2}{a^2 - c^2} = 1$$

Since $b^2 = a^2 - c^2$ (by the Pythagorean theorem):

$$\boxed{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1}$$

Standard Equations (Centre at Origin)

Horizontal Major Axis ($a > b$)

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

  • Vertices: $(\pm a, 0)$
  • Foci: $(\pm c, 0)$ where $c^2 = a^2 - b^2$
  • Major axis length: $2a$
  • Minor axis length: $2b$

Vertical Major Axis ($b > a$)

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

  • Vertices: $(0, \pm b)$
  • Foci: $(0, \pm c)$ where $c^2 = b^2 - a^2$
  • Major axis length: $2b$
  • Minor axis length: $2a$

Standard Equations (Centre at $(h,k)$)

Horizontal major axis ($a > b$): $$ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 $$

  • Vertices: $(h \pm a, k)$
  • Foci: $(h \pm c, k)$ where $c^2 = a^2 - b^2$

Vertical major axis ($b > a$): $$ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 $$

  • Vertices: $(h, k \pm b)$
  • Foci: $(h, k \pm c)$ where $c^2 = b^2 - a^2$

Key Features

  • Major axis length $2a$, minor axis length $2b$ (where $a$ is the semi-major axis)
  • Co-vertices at distance $b$ from centre (endpoints of minor axis)

Latus Rectum

The latus rectum is the chord through a focus perpendicular to the major axis.

For a horizontal ellipse, substituting $x = c$ gives $y = \pm \frac{b^2}{a}$.

Length of latus rectum: $$\boxed{\frac{2b^2}{a}}$$

(For a vertical ellipse, the length is $\frac{2a^2}{b}$.)

General Equation

Expanding the standard equation with centre $(h,k)$:

$$b^2(x-h)^2 + a^2(y-k)^2 = a^2b^2$$

$$b^2x^2 - 2hb^2x + a^2y^2 - 2ka^2y + K = 0$$

This takes the form: $$\boxed{Ax^2 + By^2 + Cx + Dy + K = 0 \quad ; \quad A \neq B}$$

where $A$ and $B$ have the same sign.

To convert from general to standard form, complete the square separately for $x$ and $y$.

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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