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?
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.