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.