I found the following problem from a book.
Let A = (-1, 0), B = (1, 0) and k = a constant which is not equal to 1. C(x, y) is a variable point such that AC = kBC. Find the locus of C.
The solution provided says something like the following:-
With the given conditions, AC = kBC means the locus of C is a circle.
Using analytic geometry to connect both sides by distance formula, I …. (finally) obtain the equation of C as
$\ (x – \dfrac {K + 1}{K - 1})^2 + y^2 = (\dfrac {K + 1}{K - 1})^2 – 1$; where $K = k^2$
The locus is indeed a circle.
My questions are:-
(1) Does a circle always have the characteristic of AC = kBC so that we can jump to the said conclusion?
(2) Can “the locus of C being a circle under the conditions that A and B are fixed points” be proved geometrically?
(3) I have tested that for some k (and probable for all k, I suspect) neither A nor B are points on the circle. Do we have special names for A and B?