# chordal

By the entry, the power of the point $(a,b)$ with respect to the circle

$${K}_{1}(x,y):={(x-{x}_{1})}^{2}+{(y-{y}_{1})}^{2}-{r}_{1}^{2}=0$$ |

is equal to ${K}_{1}(a,b)$ and with respect to the circle

$${K}_{2}(x,y):={(x-{x}_{2})}^{2}+{(y-{y}_{2})}^{2}-{r}_{2}^{2}=0$$ |

equal to ${K}_{2}(a,b)$. Thus the locus of all points $(x,y)$ having the same with respect to both circles is characterized by the equation

$${K}_{1}(x,y)={K}_{2}(x,y).$$ |

This reduces to the form

$$2({x}_{2}-{x}_{2})x+2({y}_{2}-{y}_{1})y+k=0,$$ |

and hence the locus is a straight line perpendicular^{} to the of the circles. This locus is called the chordal or the radical axis of the circles.

Title | chordal |
---|---|

Canonical name | Chordal |

Date of creation | 2013-03-22 15:07:50 |

Last modified on | 2013-03-22 15:07:50 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 9 |

Author | PrimeFan (13766) |

Entry type | Result |

Classification | msc 51M99 |

Classification | msc 51N20 |

Synonym | radical axis |