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# chordal

By the parent 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 power 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 centre line of the circles. This locus is called the chordal or the radical axis of the circles.

Synonym:

radical axis

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Reference

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## Mathematics Subject Classification

51M99*no label found*51N20

*no label found*

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