Let A and B be two points on the hyperbola xy = 1 and let C be the reflection B of through the origin.

This is a two part problem -

(a) Show that C is on the hyperbola.

(b) Let Gamma be the circumcircle of triangle ABC and let A' be the point on Gamma diametrically opposite A. Show that A' is also on the hyperbola xy=1.

(a) is fairly easy to prove, as we can see that B = (x, y), thus B reflected across the origin must be C = (-x, -y). From this, we can see -x(-y) = xy = 1.

However, I'm not too sure about (b). Could somebody help me with that?

Guest Feb 21, 2020