A circle (of radius a), a line, and a point are mapped by inversion into two concentric circles and the center of those concentric circles. If the distance from the given circle's center to the line is b, then what is the distance from the point to the line?

Inversion Defined:

Let O be the center of a circle of radius k. An inversion with respect to circle O is a mapping f:R² -> R² such that for all P in R² (not O), P' = f(P) lies on ray OP and
|OP'||OP| = k².

See www.geocities.com/bractals/inv.jpg

for graphical description of inversion.

Nice problem.  Am I reading it wrong?

Please check the position of the center of the magenta circle shown on your drawing at www.geocities.com/bractals/inv.jpg . I believe the center of the magenta circle must be located on the circumference of the circle of inversion in order to map onto itself. 

Let C be the center of the magenta circle, then |OC|*|OC'| = k² only if |OC| = k.

(Because C and C' are located at the same position.)

Edited on February 23, 2006, 9:23 pm

Posted by Mindrod on 2006-02-23 21:09:01