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

Question

(a) Show that C is on the hyperbola.

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

Answer 1

Sorry, I submitted the question before writing some things. Here's what I don't understand: what does it mean when C is the reflection of B through the origin? I thought that points can only be reflected through lines?

Answer 2

reflection through the origin takes (x,y) --> (-x,-y)
Clearly (-x)(-y) = xy
The hyperbola is symmetric about the origin.

google the topic and you will find more discussions and illustrations.

Answer 3

Thanks, I solved (a).

Answer 4

I also almost solved (b), just one thing I don't get: what do we do once we have found the coordinates of A'?

Answer 5

Oh never mind, I have finished the problem.

Answer 6

How do we find the coordinates of A'?