Curves of the second order, also known as conic sections, are curves obtained by the intersection of a plane and a double-napped cone. These include ellipses, parabolas, and hyperbolas. They are defined by quadratic equations of the form:

$$ A x^2+B x y+C y^2+D x+E y+F=0 $$

where $A, B, C, D, E, F$ are constants.

Types of Second-Order Curves

1. Ellipse (includes the Circle)

   • General equation:

   $$ \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 $$

   • If $a=b$, it is a circle.
   • If $A, C>0$ and $B^2-4 A C<0$, it's an ellipse.

2. Parabola

   • General equation:

     $$ y=a x^2+b x+c $$

   • A parabola has one focus and one directrix.

   • If $B^2-4 A C=0$, it's a parabola.

3. Hyperbola

   • General equation:

     $$ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 $$

   • A hyperbola consists of two branches.

   • If $B^2-4 A C>0$, it's a hyperbola.

Special Cases

• Degenerate cases occur when the conic reduces to a pair of lines, a point, or disappears.

🧠To plot ellipse, parabola, and hyperbola

https://gist.github.com/viadean/a5515c762b1024e489c6313653121b5c

Explanation of the Code:

1. Ellipse: Uses parametric equations $x=h+a \cos (t), y=k+b \sin (t)$.
2. Parabola: Uses standard quadratic equation $y=a(x-h)^2+k$.
3. Hyperbola: Uses hyperbolic functions $x=h \pm a \cosh (t), y=k \pm b \sinh (t)$.
4. Plots all curves on the same graph with a legend and axis labels.