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At 7:17 of this video by Eddie Woo, he states that the axis passing through focus of hyperbola is conjugate axis. If this is so, isn't the length of intercept made by this line through hyperbola, just the length of latus rectum?

If so, I can't understand the equations for length of each of these objects. The length for lactus rectum is given as $ 2a(e^2-1)$ and of conjugate axis as $2b$ for the hyperbola:

$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} =1$$ Having eccentricity $e$

tryst with freedom
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  • The conjugate axis ($2b$) and latus rectum ($2b^2/a$) are almost-never the same length. ... It looks like Woo wanted to draw a vertical line to label "conjugate axis" somewhere, and inadvertently put it confusingly-close to a focus. At that point in the lesson, Woo's being a bit vague about things in order to drive discussion about what the name even means. He may be anticipating drawing the conjugate hyperbola later, and, not wanting to give away that that hyperbola's transverse axis is the original hyperbola's conjugate, draws his "conjugate axis" indicator off to the side. – Blue Feb 24 '21 at 14:20
  • Thank you so much for clearing that up, could you give a reference to where I can study the derivation for length of conjguate axis @Blue – tryst with freedom Feb 24 '21 at 14:49
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    Older texts on conic sections tend to be pretty thorough, but I don't have a particular reference. This may help: The notion of "transverse axis" is obvious enough, like an ellipse's "major axis"; "conjugate axis" is more elusive because you can't see it. Except ... If you draw a rectangle of transverse-axis "width", whose diagonals align with the hyperbola's asymptotes, then the "height" of that rectangle matches the length of the conjugate axis; so, in a way, you can see it. (BTW: The "conjugate hyperbola" has the same asymptotes, but with vertices on the other sides of that rectangle.) – Blue Feb 24 '21 at 15:14
  • Omg it it is just the length between the vertice of conjugate hyperbola @Blue – tryst with freedom Feb 24 '21 at 15:19
  • See the last sentence of my first comment. :) – Blue Feb 24 '21 at 15:20
  • Oh lol, I missed that but I got it's all clear now :P – tryst with freedom Feb 24 '21 at 15:23

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