Conic Sections

Overview

Conic sections are curves obtained by intersecting a cone with a plane: circles, ellipses, parabolas, and hyperbolas.

Circle

Standard Form (Center at origin)

$x^2 + y^2 = r^2$

Standard Form (Center at $(h, k)$)

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

Property	Value
Center	$(h, k)$
Radius	$r$

Parabola

Vertical Axis (opens up/down)

$(x - h)^2 = 4p(y - k)$

• Vertex: $(h, k)$
• Focus: $(h, k + p)$
• Directrix: $y = k - p$
• $p > 0$: opens upward
• $p < 0$: opens downward

Horizontal Axis (opens left/right)

$(y - k)^2 = 4p(x - h)$

• Vertex: $(h, k)$
• Focus: $(h + p, k)$
• Directrix: $x = h - p$
• $p > 0$: opens right
• $p < 0$: opens left

Ellipse

Horizontal Major Axis

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

Property	Value
Center	$(h, k)$
Vertices	$(h \pm a, k)$
Co-vertices	$(h, k \pm b)$
Foci	$(h \pm c, k)$ where $c^2 = a^2 - b^2$
Major axis length	$2a$
Minor axis length	$2b$

Vertical Major Axis

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

Property	Value
Center	$(h, k)$
Vertices	$(h, k \pm a)$
Co-vertices	$(h \pm b, k)$
Foci	$(h, k \pm c)$ where $c^2 = a^2 - b^2$

Eccentricity

$e = \frac{c}{a} \quad \text{(where } 0 < e < 1 \text{ for ellipse)}$

Hyperbola

Horizontal Transverse Axis

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

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

Vertical Transverse Axis

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

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

Eccentricity

$e = \frac{c}{a} \quad \text{(where } e > 1 \text{ for hyperbola)}$

General Second-Degree Equation

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

Discriminant: $B^2 - 4AC$

Value	Conic
$B^2 - 4AC < 0$	Ellipse (or circle if $A = C$ and $B = 0$)
$B^2 - 4AC = 0$	Parabola
$B^2 - 4AC > 0$	Hyperbola

Quick Reference

Conic	Relationship	Eccentricity
Circle	$r = \text{constant}$	$e = 0$
Ellipse	$c^2 = a^2 - b^2$	$0 < e < 1$
Parabola	Vertex to focus $= p$	$e = 1$
Hyperbola	$c^2 = a^2 + b^2$	$e > 1$