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I am missing some fundamental hyperbola geometry-related concepts probably and I cannot get more insight from the websites, so I'd like to ask the following from this forum.

Let A = x(A), B = Length(B), and C = y(C)

Is there any neat way to explain, why the run and the rise line B/A (x = yB/A) intersecting hyperbola (y^2-x^2 = A^2) and the tangent of the circle with the radius A meet at the Y-coordinate on point C? Should be also said that the tangent crosses the x-axis in x=B.

I can see how the math equates, but I feel I have not understood why seemingly different geometric constructions yield the same position on the Y-axis. Is this due to some explicit mathematical theorem?

Below is the construction. Pardon me for not providing more code at this point:

Hyperbola tangent

The line and the hyperbola is solved in the picture C = AB/sqrt(B^2-A^2).

MarkokraM
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    I am sorry but your question is very unclear. You don't say how you define your hyperbola .... I don't understand what is "the line B/A" you are speaking about ? Do you mean with equation y=(B/A)x ? Moreover $C$ means two different things a point and an expression, no in fact two expressions : which one is the good one ? – Jean Marie Feb 01 '21 at 20:25
  • Thanks for the input, I tried to elaborate on various parts of the question. Let me know if there is something else required. – MarkokraM Feb 02 '21 at 05:54
  • This might relate to the two concentric circles with a tangent to the inner circle scenario. I have also found that the parabola x = y² sqrt((x(C))² - (x(B))²) gives the same focal point in Y-axis. – MarkokraM Feb 02 '21 at 19:27
  • Moreover, I think, this is what I was looking for, hyperbolic functions and their relation to trigonometric counterparts: https://math.stackexchange.com/questions/451034/geometric-construction-of-hyperbolic-trigonometric-functions – MarkokraM Feb 03 '21 at 16:14

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