rectangular hyperbola chords part 1

by OronSH, Apr 9, 2024, 8:01 PM

Let $\mathcal{H}$ be a rectangular hyperbola with center $Z$ , and let $\overline{PQ}$ be a chord of $\mathcal{H}$ with midpoint $M$ . Prove that:
The pole of $\overline{PQ}$ with respect to $\mathcal{H}$ lies on line $ZM$ .

Since $(P,Q;M,\infty)=-1$ the polar of $M$ passes through the point at infinity along $PQ,$ and pole polar duality shows that the line through $M$ and the pole of $PQ$ is the polar of a point at infinity, which clearly must pass through $Z.$

1 xook
susus

Comment

1 Comment

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

this is true for all conics its written in the hyperbola unit like this wait for part 2

This post has been edited 1 time. Last edited by OronSH, Apr 10, 2024, 3:01 PM

by DottedCaculator, Apr 10, 2024, 12:47 AM

Report

Submit

oron the rizzler admits 10 quandillion aura, oron so orz
by ohiorizzler1434, Apr 24, 2025, 12:29 AM
I love that I am banned
by maxamc, Apr 23, 2025, 2:59 AM
because it was SPAM
by PEKKA, Apr 18, 2025, 6:23 PM
wait why was a post by PEKKA deleted lol
by lpieleanu, Apr 17, 2025, 7:21 PM
Hi! contrib pls? I literally have a math team named xooks lol
by Cattus1221, Apr 16, 2025, 9:45 PM
sus_rbo.
by sus_rbo, Apr 11, 2025, 12:06 PM
wsg my second favorite borosillicate glass boron isotope orzon xoron oreo
by Inaaya, Mar 30, 2025, 10:38 PM
wsg my second favorite borosillicate glass boron isotope orzon xoron oreo
by Pengu14, Mar 30, 2025, 2:51 AM
wsg my favorite borosillicate glass boron isotope orzon xoron oreo
by Inaaya, Mar 30, 2025, 2:14 AM
ORZ ORZ ORZ xooks
by Mr.Sharkman, Mar 28, 2025, 12:39 AM
ohiorizzler1434
by sixoneeight, Mar 23, 2025, 8:19 PM
exiters!!! xooks!! admits!!!!! now that's rizz!!!! xioooixiooix xoink!!!!
by babyzombievillager, Mar 22, 2025, 1:40 AM
exiters!!! xooks!! admits!!!!! now that's rizz!!!! xioooixiooix xoink!!!!
by Pengu14, Mar 21, 2025, 9:43 AM
exiters!!! xooks!! admits!!!!! now that's rizz!!!! xioooixiooix xoink!!!!
by bjump, Mar 18, 2025, 5:04 PM
exiters!!! xooks!! admits!!!!! now that's rizz!!!! xioooixiooix xoink!!!!
by ohiorizzler1434, Mar 18, 2025, 1:40 AM
or on rbo so xor susus
by Orthogonal., May 17, 2023, 5:00 AM
xooks rbos .
by OronSH, May 17, 2023, 3:13 AM
sus us. $\text{[math]}$
by mathfan2020, May 17, 2023, 3:01 AM
how so xor oren W
by peelybonehead, May 17, 2023, 2:49 AM
xook $\text{[math]}$ $\text{[math]}$
by OronSH, May 17, 2023, 2:44 AM
Hihihihihihihihi
by maxamc, May 17, 2023, 2:27 AM
hi orons !
by megarnie, May 17, 2023, 2:12 AM
ooks . $~~~~~~~$
by pi271828, May 17, 2023, 2:10 AM
how so orz
by UnearthedCyclone, May 17, 2023, 2:09 AM
so orz

by zhoujef000, May 17, 2023, 1:45 AM
imagine so xor
by pikapika007, May 17, 2023, 1:45 AM
XORON HOWSOXOR
by GoodMorning, May 17, 2023, 1:45 AM
HOW SO XOR
by TotallyNotSimon, May 17, 2023, 1:44 AM
orz $\text{[math]}$
by aa1024, May 17, 2023, 1:44 AM

413 shouts

xooks

rectangular hyperbola chords part 1

by OronSH, Apr 9, 2024, 8:01 PM

1 Comment