--- In Hyacinthos@yahoogroups.com
, "Francisco Javier" <garciacapitan@...> wrote:
> Dear friends:
> I now see that this is a particular case of some projective theorem: "the locus of poles with respect a conic of the tangents to another conic is also a conic".
> Here is the version for two circles:
> (A) and (B) are circles
> The line AB intersect (B) at M and N
> M' and N' are the inverses of M and N with respect to (A)
> J is the inverse of A with respect to (B)
> O is the inverse of J with respect to (A)
> A' is the reflection of A on O
> The locus points P such that the polar of P with respect to (A) is tangent to (B) is a conic with foci A and A' and diameter M'N'.
> what is the description of the locus in the general case in terms of the two given conics?
> Thank you.
Dear Francisco, it is the polar conic of one conic with respect to another, in other words, the dual of a conic is a conic. For further properties you can see Geometry of conics by A.V. Akopyan and A.A. Zaslavsky, pages 70-72.