Ellipses and Hyperbolas

Ellipses

An ellipse is the set of all points where the sum of distances to two foci is constant.

Standard Form (Horizontal Major Axis)

$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1, \quad a > b$

Standard Form (Vertical Major Axis)

$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1, \quad a > b$

Where:

• $(h, k)$ = center
• $a$ = semi-major axis (larger value)
• $b$ = semi-minor axis (smaller value)
• $c$ = distance from center to focus

Key relationship: $c^2 = a^2 - b^2$

Key Features of Ellipse

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

• Center: $(h, k)$
• Vertices (endpoints of major axis): $(h \pm a, k)$
• Co-vertices (endpoints of minor axis): $(h, k \pm b)$
• Foci: $(h \pm c, k)$ where $c^2 = a^2 - b^2$
• Major axis length: $2a$
• Minor axis length: $2b$

For vertical major axis: Swap the roles (vertices on vertical axis)

Eccentricity

$e = \frac{c}{a}, \quad 0 < e < 1$

• $e$ close to $0$: nearly circular
• $e$ close to $1$: very elongated

Hyperbolas

A hyperbola is the set of all points where the difference of distances to two foci is constant.

Standard Form (Horizontal Transverse Axis)

$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

Opens left and right.

Standard Form (Vertical Transverse Axis)

$\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$

Opens up and down.

Where:

• $(h, k)$ = center
• $a$ = distance from center to vertex
• $b$ = determines spread of branches
• $c$ = distance from center to focus

Key relationship: $c^2 = a^2 + b^2$ (note: plus, not minus!)

Key Features of Hyperbola

For horizontal transverse axis $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$:

• Center: $(h, k)$
• Vertices: $(h \pm a, k)$
• Foci: $(h \pm c, k)$ where $c^2 = a^2 + b^2$
• Asymptotes: $y - k = \pm\frac{b}{a}(x - h)$

For vertical transverse axis: Vertices and foci on vertical axis, asymptotes: $y - k = \pm\frac{a}{b}(x - h)$

Asymptotes

The branches of a hyperbola approach (but never touch) the asymptotes.

Rectangle method:

1. Draw rectangle with vertices at $(h \pm a, k \pm b)$
2. Draw diagonals of rectangle
3. These diagonals are the asymptotes

Identifying Conic Sections

From the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ (with $B = 0$):

• Circle: $A = C$ (and same sign)
• Ellipse: $A \neq C$ but same sign
• Parabola: Either $A = 0$ or $C = 0$ (but not both)
• Hyperbola: $A$ and $C$ have opposite signs

Summary Table

| Conic | Standard Form | $c$ relationship | Key feature | |-------|---------------|------------------|-------------| | Circle | $(x-h)^2 + (y-k)^2 = r^2$ | N/A | All points distance $r$ from center | | Ellipse | $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ | $c^2 = a^2 - b^2$ | Sum of distances to foci = $2a$ | | Hyperbola | $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ | $c^2 = a^2 + b^2$ | Difference of distances to foci = $2a$ | | Parabola | $(x-h)^2 = 4p(y-k)$ | N/A | Distance to focus = distance to directrix |