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[parent] chordal (Result)

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

$\displaystyle K_1(x,\,y) := (x-x_1)^2+(y-y_1)^2-r_1^2 =0$
is equal to $ K_1(a,\,b)$ and with respect to the circle
$\displaystyle K_2(x,\,y) := (x-x_2)^2+(y-y_2)^2-r_2^2 =0$
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
$\displaystyle K_1(x,\,y) = K_2(x,\,y).$
This reduces to the form
$\displaystyle 2(x_2-x_2)x+2(y_2-y_1)y+k = 0,$
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.



"chordal" is owned by PrimeFan. [ owner history (1) ]
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Other names:  radical axis

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Cross-references: perpendicular, line, straight, equation, points, locus, circle, power of the point
There is 1 reference to this entry.

This is version 6 of chordal, born on 2005-03-11, modified 2005-03-12.
Object id is 6873, canonical name is Chordal.
Accessed 2132 times total.

Classification:
AMS MSC: 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
 51M99 (Geometry :: Real and complex geometry :: Miscellaneous)
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