# Conic Sections

rename
Updated
2009-02-03 09:53

## Conic Sections

In general, Conic Sections satisfy: Ax^{2}+Bxy+Cy

^{2}+Dx+Ey=F.

Conic Section | Equation |
---|---|

Upward-opening parabola with vertex h,k | (x-h)^{2} = 4c(y-k), c>0 |

Downward-opening parabola with vertex h,k | (x-h)^{2} = 4c(y-k), c<0 |

Rightward-opening parabola with vertex h,k | (y-k)^{2} = 4c(x-h), c>0 |

Leftward-opening parabola with vertex h,k | (y-k)^{2} = 4c(x-h), c<0 |

Circle with center h,k | (x-h)^{2} + (y-k)^{2} = r^{2} |

Ellipse with x-interecepts h+-a and y-intercepts k+-b | (x-h)^{2}/a^{2} + (y-k)^{2}/b^{2} = 1 |

Hyperbola with x-intercepts h+-a | (x-h)^{2}/a^{2} - (y-k)^{2}/b^{2} = 1 |

Hyperbola with y-intercepts k+-b | (y-k)^{2}/b^{2} - (x-h)^{2}/a^{2} = 1 |

c=distance of foci from h,k, the center of an ellipse | c=sqrt(|a^{2}-b^{2}|) |

c=distance of foci from h,k, the center of a hyperbola | c=sqrt(a^{2}+b^{2}) |

Conic Section | Eccentricity e |
---|---|

Circle | e=0 |

Parabola | e=1 |

Ellipse | e=c/a, 0 |

Hyperbola | e=c/a, e>1 |