Intersecting Chord Theorem for Conic Sections

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: $1 \text b$.