Intuition of the equation of a hyperbola

Question

The standard form of a circle is:
${x^2} + {y^2}={r^2}$
Which shows that the circle is a locus of points with the same distance from the focus (centre).

The standard form of an ellipse is:
$\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$
Which shows that the ellipse is a circle stretched horizontally by a factor of a, and stretched vertically by a factor of b.

The standard form of an hyperbola is:
$\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$
Which I fail to understand intuitively.

Why does swapping the sign from + to - turn an ellipse into a very different-looking hyperbola? Is there a short intuitive reason as to why?

I've been searching for answers, and came across a thread that mentioned that for an ellipse, $y\to iy$ will cause it to become a hyperbola.
It was mentioned that "the hyperbola is the family of curves of orthogonal trajectory to an ellipse".

Can someone simplify this or explain it? I know that multiplying by $i$ will cause a 90 degree rotation anticlockwise on a complex plane. Since an ellipse is on an xy-plane, does that mean that you can add an imaginary z-axis and the ellipse can somehow become a hyperbola? This is very difficult to visualise.

Answer 1

Consider the expression$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$$If it holds for a pair $(x,y)$, then $\lvert y\rvert$ cannot be very large. In fact, we can't have $\lvert y\rvert>\lvert b\rvert$, because then $\frac{x^2}{a^2}<0$, which is impossible.

But if you consider the expession$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$$then there os no restricton on $y$. No matter how large it is, you can always take$$x=\pm a\sqrt{1+\frac{y^2}{b^2}}.$$