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I've a kind of strange doubt. I was doing some exercises about conic sections and then I realized that a hyperbola and an ellipse have some strange relation.

If we have the ellipse equation

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

and we make the transformation $b\to bi$ (where $i^2=-1$), then we have now a hyperbola. I've done some problems and derived some formulas using that fact.

Therefore, I've a conceptual doubt now. Is there a deep meaning in this transformation, or is it just a coincidence?

Ted Shifrin
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Matheus Domingos
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  • I think you meant hyperbola. – Steven Alexis Gregory Aug 26 '18 at 03:10
  • Oh sorry, yes, I'll correct it – Matheus Domingos Aug 26 '18 at 03:13
  • $b$ is an arbitrary constant, changing $b$ gives another conic section. Probably transforming $y\to iy$ can have some interpretation! – tarit goswami Aug 26 '18 at 03:13
  • But what is the interpretation. I've read that a hyperbola is like an elipse with focus on infinity. Can it be the reason? – Matheus Domingos Aug 26 '18 at 03:15
  • This notion is echoed in the way regular (circular) trig functions and hyperbolic trig functions, which can be defined using the "unit" ellipses and hyperbolas ($x^2+y^2=1$ and $x^2-y^2=1$), have expressions in terms of the exponential function. $$\cos x = \frac12\left( e^{ix} + e^{-ix}\right) \qquad ;\sin x = \frac{1}{2i}\left(e^{ix}-e^{-ix}\right)$$ $$\cosh x = \frac12 \left( e^x + e^{-x}\right) \qquad \sinh x = \frac12\left( e^x - e^{-x}\right)$$ (Interesting that the circular functions are the ones that incorporate the imaginary unit.) – Blue Aug 26 '18 at 03:17
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    @MatheusDomingos: A hyperbola is not an ellipse with "focus on infinity". A parabola can be thought of that way. You get a hyperbola by sending one of the foci of an ellipse off to infinity (getting the parabola) and then bringing that focus back from the other direction. :) You can visualize this with the classical sense of things: Let a plane intersect a double-cone, then vary the angle that the plane makes; the curve formed will change from ellipse (when the plane is less steep than the sides of the cone) to parabola (when steepness matches) to hyperbola (when the plane is steeper). – Blue Aug 26 '18 at 03:19
  • Ohh that's true. This is a cool fact, good. Does it have some useful geometric interpretation? – Matheus Domingos Aug 26 '18 at 03:19
  • What do you mean by bringing that focus from the other direction? I haven't understand it well – Matheus Domingos Aug 26 '18 at 03:22
  • @MatheusDomingos: See this answer. The answer's animation stops at the parabola step, but the animation linked in the "indeed" comment shows the transition to hyperbola as the traveling focus comes back around. – Blue Aug 26 '18 at 03:26
  • @MatheusDomingos See this answer for some geometric insights. – dxiv Aug 26 '18 at 03:28
  • Does this answer your question? Hyperbola: A case of an ellipse? – Alex M. Jun 17 '23 at 08:19

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Transforming $b\to bi$ just gives an equation of another conic section which is a hyperbola (as $b$ is an arbitrary variable, so transforming it doesn't make sense). Actually transforming $y\to iy$ has some meaning. Multiplying by $i$ means rotating the line, joining the complex number and origin, by $90^{\circ}$ keeping the value same( as $|i|=1$). Gives the result that, family of curves which are orthogonal trajectory to an ellipse which has the equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ is the hyperbola.

tarit goswami
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