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 $Z.$

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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

susus

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