Common second-order conclusions of conic curves:

1, ellipse: focus radius: a+ex (left focus), a-ex (right focus), x=a? /c .

2. hyperbola: focal radius: | a+ex | (left focal point) |a-ex| (right focal point), directrix x=a? /c .

3. parabola (y? = 2px): focal radius: x+p/2 directrix: x =-p/2.

Extended knowledge

1. What is a conic curve?

A conic curve is a curve obtained by cutting a plane into a conic surface. Conic curves include ellipse (circle is a special case of ellipse), parabola and hyperbola. The ancient Greek mathematicians who originated more than 2000 years ago first began to study conic curves.

The (incomplete) unified definition of conic: the locus of a point whose ratio of the distance r from a point to a point on a plane to the distance d from a point to a straight line is constant e=r/d is called conic. When e> 1 is hyperbola, when e= 1 is parabola, when 0

The fixed point is called the focus of the conic, the fixed line is called the directrix, and e is called the eccentricity.

2. Origin

More than 2000 years ago, ancient Greek mathematicians first began to study conic curves, and achieved a lot of results. Apollonius, an ancient Greek mathematician, studied these curves with the method of plane truncated cone.

Cut the cone with a plane perpendicular to the axis of the cone and you get a circle; Tilt the plane gradually to get an ellipse; When the plane is inclined to "and only parallel to a generatrix of the cone", a parabola is obtained; When the plane parallel to the axis of the cone is cut, a hyperbola can be obtained (when the conical surface is replaced by the corresponding conical surface, a hyperbola can be obtained).

Apollo once called ellipse "deficient curve", hyperbola "hypercurve" and parabola "homogeneous curve". In fact, in his works, Apolloni has obtained all the properties and results of conic curves in today's high school mathematics by means of pure geometry.