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DoeADeer

Member

Registered: 2019-10-18

Posts: 2

Analytic Geometry- PLEASE HELP!

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

(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.

I don't know where to start. I tried graphing it but it doesn't make any sense to me. Thanks for any help!

DoeADeer

Member

Registered: 2019-10-18

Posts: 2

Re: Analytic Geometry- PLEASE HELP!

Never mind, I have completed the problem.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Analytic Geometry- PLEASE HELP!

hi DoeADeer

Welcome to the forum.

Thanks for posting anyway.  It was an interesting problem to have a go at.  First part was fairly quick to do.  I found the second part tricky as it took many steps.

My method was

(1) Write down the equation for the perpendicular bisector of CB.
(2) Calculate the equation for the perpendicular bisector of AB.
(3) Use these to find the centre D of the circumcircle.
(4) Get the vector AD and use this to get the coordinates of A'. (AD = DA')
(5) Show those coordinates satisfy the equation of the hyperbola.

Do you have a simpler method?  I'd like to see it if you do.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob