Chordal 2005-03-11 12:50:00 2005-03-12 06:08:54 Result power of point % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here By the \PMlinkescapetext{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 \PMlinkescapetext{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 \PMlinkescapetext{{\em centre line}} of the circles. \,This locus is called the {\em chordal} or the {\em radical axis} of the circles.