radical axis

[AntonioLao]

Oxford’s concise dictionary of mathematics defines radical axis of two circles as the straight line containing all points P such that the lengths of the tangents from P to the two circles are equal. However, if the circles intersect in two points then the straight line passing thru them is the radical axis and the common chord contains a subset of P of infinite points from which tangents to the two circles cannot be drawn. If the two circles are tangent at a point p of P then all points of P are on the tangent line. If the two circles do not intersect then the radical axis contains imaginary solutions extended into the complex plane.

In the study of analytic geometry of 2 dimensional Cartesian coordinate system where x=0 is the ordinate vertical axis and y=0 is the abscissa horizontal axis, the general equations of the two circles are x+y+2ax+2by+c=0 and x+y+2dx+2ey+f=0 where a, b, c, d, e, f are all real numbers then the radical axis has the linear equation: 2(a-d)x+2(b-e)y+(c-f)=0 whose slope is given by (d-a)/(b-e) and y-intercept is given by ½(f-c)/(b-e). The sufficient condition for the radical axis to become a vertical line is b=e and if c=f then it is the vertical line x=0. The sufficient condition for the radical axis to become a horizontal line is a=d and if c=f then it is the horizontal line y=0. However, if both conditions a=d and b=e are satisfied then the radical axis vanishes indicating that the circles are concentric.

[mkirkpatrick]

I would prefer a non-radical axis personally!



regards michael.

[AntonioLao]

A non-radical axis is the same as an imaginary axis where circles of influence never intersect with each other as can be found in the quantum vacuum of spacetime.

[mkirkpatrick]

A non-radical axis is the same as an imaginary axis where circles of influence never intersect with each other as can be found in the quantum vacuum of spacetime.

Just where would we be without the vacuum?




regards michael.

[AntonioLao]

Only the quantum vacuum has any bearing for the formations of mass and energy. The continuous vacuum is non-reactive and non-participatory. It is the absolute passive observer of absolute consciousness.