Intersecting Chord Theorem for Conic Sections
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Theorem
Consider a right circular cone $\CC$ with opening angle $2 \alpha$ whose apex is at $O$.
Consider a slicing plane $\PP$, not passing through $O$, at an angle $\beta$ to the axis of $\CC$.
Let the plane $OAA'$ through the axis of $\CC$ perpendicular to $\PP$ intersect $\PP$ in the line $AA'$.
Let $P$ be an arbitrary point on the intersection of $\PP$ with $\CC$.
Let $PM$ be constructed perpendicular to $AA'$.
Then:
- $PM^2 = k \cdot AM \cdot MA'$
where $k$ is the constant:
- $k = \dfrac {\map \sin {\beta + \alpha} \map \sin {\beta - \alpha} } {\cos \alpha}$
Proof
This theorem requires a proof. In particular: This is part of the muddled Construction of Conic Section in Sommerville. This collection of pages needs to be revisited with respect to a more coherent work than this one. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1 \text b$.