The (incomplete) unified definition of conic: the locus of a point whose quotient of the distance to a fixed point (focus) and the distance to a fixed line (directrix) is constant E (eccentricity). When e> is 1, it is a hyperbola; when e= 1, it is a parabola; when 0
More than 2000 years ago, ancient Greek mathematicians first began to study conic curves [1-3]? , 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.
Geometric viewpoint
A conic section is obtained by cutting a conic surface with a plane.
Generally speaking, conic curve includes ellipse, hyperbola and parabola, but strictly speaking, it also includes some degenerate cases. Specifically:
1) When the plane is parallel to the generatrix of the quadric surface and does not pass through the vertex of the quadric surface, the result is a parabola.
2) When the plane is parallel to the generatrix of the quadric surface and passes through the vertex of the quadric surface, the result degenerates into a straight line.
3) When the plane only intersects one side of the conic surface and does not pass through the vertex of the conic surface, the result is an ellipse.
4) When the plane intersects only one side of the conical surface, but not the apex of the cone, and is perpendicular to the axis of symmetry of the cone, the result is a circle.
5) When the plane intersects only one side of the conic surface and passes through the vertex of the conic, the result is a point.
6) When the plane intersects both sides of the quadric surface and does not pass through the vertex of the quadric surface, the result is a hyperbola (each branch is the intersection of a quadric surface in this quadric surface and the plane).
7) When the plane intersects with both sides of the quadric surface and passes through the vertex of the quadric surface, the result is two intersecting straight lines.
Note that the above curve class does not contain quadratic curves: two parallel straight lines.
Algebraic viewpoint
On the Cartesian plane, the binary quadratic equation
The image is called a conic. According to the different discriminant, it includes ellipse, hyperbola, parabola and various degradation situations.
Focus directrix and its extended viewpoint
Traditional unified definition of focus directrix
(The focus-directrix viewpoint used for many years can only define the main situation of conic curve, and cannot be regarded as the definition of conic curve. However, because of its concise and beautiful form and many important geometric concepts and properties in conic curves, it is favored and widely used. )
Given a point P, a straight line L and a non-negative real constant E, the locus of the point whose distance to P is the ratio of the distance to L is a conic curve.
According to the range of e, the curves are different. Details are as follows:
1) e=0, and the trajectory is a point or a circle;
2) e= 1 (that is, the distance from p to l is the same), and the trajectory is parabolic;
3)0 & lt； E< 1, the trajectory is ellipse;
4)e & gt； 1, the trajectory is hyperbola.
2. Unified definition of the first and second curves.
(In Mathematics Bulletin 20 16. 12, the paper "The Trajectory Unity and Properties of the First and Second Curves", Hu Xinping, a domestic middle school math teacher, popularized the focus-directrix, thus giving the following complete and unified definitions of the first and second curves)
There are two straight lines l and m perpendicular to each other and intersect at point e on the plane. Point F is a point on a straight line M, |EF|=p, and point N is a fixed point on a straight line L. The trajectory moving point M satisfies the following two conditions at the same time:
(I) the directed distances Nm and Mm from the moving point n and the moving point m to the fixed straight line m are
Picture 3
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Nm=( 1+t)Mm, where t is a real constant;
(ii) The distance from the moving point M to the fixed point F |MF| and the distance from the moving point N |MN| are
|MF|=e|MN|, where e is a nonnegative constant,
Then from the point of view of rectangular coordinate transformation, the trajectory of moving point M is a quadratic curve.
(Conventions e= 1, t = 1 and p=0 are not valid at the same time).
The trajectory of point m is as follows:
(a) When p ≠ 0: There are six kinds of linear curves and quadratic curves.
E≠0，
(1) When e= 1, |t|= 1, the trajectory is a single straight line;
(2) When e= 1, |t|≠ 1, the trajectory is a parabola;
Figure 1
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(3) When E
(4) When e≠ 1 and e|t|= 1, the trajectory is two parallel straight lines;
(5) when e
When e=0, the trajectory is one point.
(b) When p = 0, there are three kinds of linear and quadratic curves.
(1) When e
(2) When e= 1, e|t|≠ 1, or e≠ 1, e|t|= 1, the trajectory is two coincident straight lines;
(3) When E
The fixed point F and the fixed line L are called the quasi-focus of the corresponding trajectory curve and the quasi-straight line corresponding to the quasi-focus F, respectively.
(The following describes the general concepts and properties of the principal conic curve in a purely geometric way. Because most attributes are defined from the perspective of focusing directrix, some concepts may not be applicable to more general degradation cases. )
The definition of conic curve from the point of view of focal line. The fixed point mentioned in the definition is called the focus of conic section; A fixed straight line is called the directrix of a conic curve; A fixed constant (that is, the ratio of the distance from a point to the focus on the conic to the directrix) is called the eccentricity of the conic; The distance from the focus to the directrix is called the focal length; The line segment from the focal point to a point on the curve is called the focal radius. A straight line passing through the focus and parallel to the directrix intersects the conic at two points, and the line segment between these two points is called the path of the conic, which is also called the positive focal chord in physics.
Conic curves are smooth, so there are concepts of tangent and normal.
Similar to a circle, the line segment between two intersections on a straight line intersecting a conic curve is called a chord; The string passing through the focal point is called focus chord.
For the same ellipse or hyperbola, there are two combinations of "focus directrix" to get it. Therefore, an ellipse and a hyperbola have two focuses and two directrix. A parabola has only one focus and one directrix.
A conic curve is symmetrical about a straight line whose focus is perpendicular to the directrix. In the case of ellipse and hyperbola, a straight line passes through two focal points, which is called the focal axis of conic curve. For ellipses and hyperbolas, it is also about the midline symmetry of the focal line.
Pappus theorem: the focal length of a point on a conic curve is equal to the distance from the point to the corresponding directrix multiplied by eccentricity.
Pascal's Theorem: Conic curve is inscribed with hexagon. If the opposite sides are not parallel, the intersection of the extension lines of the opposite sides of a hexagon is a line. (This also applies to the case of degradation)
Brian Xiong Theorem: The circumscribed hexagon of a conic curve has three diagonal points.
The unified equations and properties of the first and second curves can be found in the article "Unity and Properties of Trajectories of the First and Second Curves" in Mathematical Bulletin 20 16 and 12.
oblong
Literal language definition: the ratio of the distance between a moving point, a fixed point and a straight line in a plane is a normal number e less than 1. The sum of the distances between a fixed point and two fixed points (focal points) on a plane is a point set with a fixed length of 2a (let the fixed point be p and the two fixed points be F 1 and F2, then PF 1+PF2=2a). The fixed point is the focus of the ellipse, the fixed line is the directrix of the ellipse, and the constant e is the eccentricity of the ellipse.
(θ is the parameter, 0≤θ≤2π)
Hyperbolic (branch of)
Literal language definition: the ratio of the distance between a moving point, a fixed point and a straight line in a plane is a constant e greater than 1; The distance difference between a moving point and two fixed points (focal points) on the plane is equal to a point set with a fixed length of 2a (let the moving point be p and the two fixed points be F 1 and F2, then │PF 1-PF2│=2a). The fixed point is the focus of hyperbola, the fixed line is the directrix of hyperbola, and the constant e is the eccentricity of hyperbola.
oblong
The light emitted from one focus of the ellipse is reflected by the ellipse, and the reflected light converges to the other focus of the ellipse. An ellipse is equivalent to a convex lens.
hyperbola
The light emitted from one focus of the hyperbola is reflected by the hyperbola, and the reverse extension line of the reflected light converges to the other focus of the hyperbola. A hyperbola is equivalent to a concave lens.
parabola
The light emitted by the focus of parabola is reflected by parabola, and the reflected light is parallel to the axis of symmetry of parabola.
A beam of parallel light is perpendicular to the directrix of the parabola and enters the opening of the parabola. After being reflected by a parabola, the reflected light converges on the focus of the parabola. Parabola is equivalent to concave mirror.
I hope it can help you solve the problem.