how are conic section curves derived?
The conic sections can be defined in several ways.
(1) In synthetic geometry, they are formed by passing a plane through a right conical surface. Imagine two cones stuck together at the apex. A plane passing through the point of intersection creates a point, a degenerate conic section. A plane tangent to the surface along a line creates a line, another degenerate conic. A plane passing perpendicular to the axis of the cone cuts a circle; a form of ellipse. A plane passing obliquely through one of the cones yields an ellipse, while a plane passing through two cones yields a hyperbola.
(2) In algebra, the equation `Ax^2+Bx+Cy^2+Dy+Exy+F=0` generates the conics; depending on the values of the coefficients.
(3) An anlytic/geometric definition is the locus of points whose distances from a fixed point (the focus) and a given line (the directrix) is a fixed ratio.