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Given a parabola of the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, where $B^2 - 4AC = 0$, what is the formula for the focal length?

Him
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    I don’t know of any formula per se, but the focal length is half the radius of curvature at the vertex. It’s not terribly difficult to find the parabola’s vertex from the general conic equation. – amd Oct 04 '18 at 21:47
  • Do you know of a convenient parameterization of the above equation for calculating radius of curvature? It seems that the vertex might also be the point where the radius of curvature is at a maximum, so that could help. – Him Oct 05 '18 at 13:40
  • With two points and the tangents at those points you can construct a quadratic Bézier parameterization. However, in digging up references for this, I discovered that I’d already answered your question, but completely forgotten about it! The second half of this answer derives a formula for the latus rectum length, which is four times the distance between the focus and vertex, from the general conic equation. – amd Oct 05 '18 at 20:05
  • Finding the vertex by looking for maximum curvature is an interesting idea. I’ll have to play with it and see if it’s any easier than other methods. – amd Oct 05 '18 at 20:06
  • Note to voters: I’m flagging this as a duplicate because the second half of one of the answers to https://math.stackexchange.com/q/2040306/265466 derives a formula from the general conic equation, too. – amd Oct 05 '18 at 20:07
  • @amd, perhaps the second half of the answer you cite deserves its own question? It is a good answer, but SE users may find it easier to find if the question more directly asked the thing that it answers. If that makes sense. :) – Him Oct 08 '18 at 15:52
  • Perhaps so. You can always ask for the question to be reopened. – amd Oct 08 '18 at 21:05

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