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.

(In reply to Solution: I think I have it right now. by Mindrod)

I really don't mean to beat you up or anything, and I think your proof is much better than mine but I can't help correcting you when you say: "The inverted image has a center located at c' =k²/(c² -a²)" when I'm sure you mean:

The inverted image has a center located at c' =ck²/(c² -a²)

Which is funny because later when you say "k²/(c²-a²) = k²/2(c-b). This simplifies to c² -2bc + a² = 0." you probably mean:

ck²/(c²-a²) = k²/2(c-b). This simplifies to c² -2bc + a² = 0. Which would now be a true statement.

Edited on February 26, 2006, 4:12 am

Posted by Eric on 2006-02-26 04:11:53