A conic curve is a curve obtained by cutting a plane into a conic surface.Conics includeellipse(Special case where circle is ellipse)parabola、hyperbola。Originated more than 2000 years ago, ancient Greek mathematicians first began to study conic curves.
The (incomplete) unified definition of conic curve (conic curve): the locus of a point whose distance r from a certain point in the plane and distance d from a fixed straight line is constant e=r/d is called a conic curve.Where, when e>1hyperbola, when e=1parabola, when 0<e<1ellipse。[6]
The fixed point is called the focus of the conic curve, the fixed line is called the guide line (corresponding to the focus), and e is calledEccentricity。
Chinese name
Conic curve
Foreign name
conic section
Definition
It is obtained by cutting the conic surface of a planecurve
More than 2000 years ago, ancient Greek mathematicians first began to study conic curves[1-3], and a lot of results have been obtained.Greek mathematicians ApollonisThe method of plane cutting cone is used to study these curves.Cut with a plane perpendicular to the cone axiscone, the result iscircular；Gradually tilt the plane to getellipse；When the plane is inclined to be parallel to one generatrix of "and only" coneparabola；A hyperbola can be obtained by intercepting the plane parallel to the axis of the cone (when the cone surface is replaced with the corresponding conic surface, it can be obtainedhyperbola）。
ApolloniThe ellipse was called "deficient curve", the hyperbola was called "hypercurve", and the parabola was called "homogeneous curve".In fact, Apolloni has obtained all the properties and results of conic curve in today's high school mathematics by using pure geometric methods in his works.
history
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There are different opinions about the first discovery of conic curve.It is said that when solving the problem of "cubic product", ancient Greek mathematicians discovered the conic curve: let x and y be a and 2aProportional middle term, i.e
, then
，
，
, so as to obtain
。It is also said that the ancient Greek mathematicians found the same result as the "cubic product" when they studied the intersection of the plane and the cone.It is also believed that ancient astronomers were makingsundialThe conic curve was found at.A sundial is a disc placed obliquely, with a pole in the center perpendicular to the surface of the disc.When the sun shines on the sundial, the movement of the pole shadow can be timed.At different latitudes, the pole tips are drawn into different conic curves.However, the invention of the sundial was lost in ancient times.
The most outstanding achievements in the early systematic research on conic curve can be said to be the ancient Greek mathematiciansApolloni（Apollonius, front 262~front 190).He andEuclidHe was a contemporary, whose great work Conic Curve andEuclid"The Origin of Geometry" is also known as the culmination of ancient Greek geometry.
In "Conic Curve", Apolloni summarized the predecessors（PlatonismMenekemoseMultiple cube problemAnd found the work of conic curve, especially Euclid's work, and carried out the work of removing the rough and storing the fine, summarizing and refining the achievements of predecessors and making them systematic. On this basis, he also put forward many ideas of his own.There are 8 chapters in the book, with a total of 487 propositions, which have completely covered the nature of conic curve, so that future scholars have almost no room to get involved for more than 1000 years.
As we all know, cutting a biconical cone with a plane will result in circles, ellipses, parabolas, hyperbolas and their degenerate forms: two intersecting lines, a line and a point.
ellipse
Here, we only introduce Apolloni's definition of conic curve.Given a circle BC and a point A outside its plane, a straight line passing through A and moving along the circumference of the circle generates a double cone.A is called the vertex of the cone, the circle is called the bottom of the cone, and the straight line from A to the center of the circle is called the axis of the cone. The axis may not be perpendicular to the bottom.
Let a section of the cone intersect the bottom at the straight line TF, take a diameter BC of the bottom circle perpendicular to TF, and the straight lines BC and TF intersect at G, so △ ABC containing the cone axis AS is called the axis triangle.For ellipses and hyperbolas, AB and AC on both sides of the axis triangle intersect the section at E and D respectively (in the case of hyperbolas, D is on the opposite extension line of AC).For parabola, one side AB of the axis triangle intersects the section at E, and the other side AC is parallel to the section, so there is no intersection point.
Take a point L on the conic curve, make LM ‖ TF through L, and intersect EG at M.Make AK ‖ EG on the plane where the axis triangle is located, and intersect the straight line BC with K.Then make EH ∨ EG.For ellipse and hyperbola, H shall meet
, while parabola meets
。Connect DH through M
, straight line DH and straight line MN intersect at X.Then cross X and make XO ∨ straight line EH at O, then for ellipse and hyperbola
, for parabola
These are two conclusions that can be proved.
In these two conclusions, ML is called a vertical marking line of the conic curve, so the conclusion shows that the area of a square with the vertical marking line as the side length is equal to the area of a rectangle with EM as the side.For ellipse, EOEH, rectangular EOXM exceeds rectangular EHNM;The parabola, EO=EH, and the rectangle EOXM just fills the rectangle EHNM.Therefore, the original names of ellipse, hyperbola and parabola are respectively "deficient curve", "hypercurve" and "homogeneous curve".This is the definition of conic curve introduced by Apolloni.
The two conclusions given by Apolloni are also easy to useMathematical symbolTo represent.Let ML=y, EM=x, ED=a, EH=p (for a given figure, a and p are fixed values), it can be proved that the ellipse satisfies
, hyperbola meets
, while the parabola satisfies
。
Note: if the section is parallel to AC, it can be considered that the section intersects AC atInfinity point, where ED=a=∞, so p/a=0, that is, the parabola can be regarded as an ellipse or a hyperbolalimitForm.
In the 13 centuries after the publication of Apolloni's "Conic Curve", there has been no new progress in the research of conic curve in the whole mathematical world.In the 11th century, Arab mathematicians used conic curves to solve cubic problemsalgebraic equationFrom the 12th century, conic curve was introduced into Europe through Arabia, but there was still no research on conic curve at that timeBreach。Until the 16th century, two things prompted people to further study conic curves.First, German astronomersKepler（Kepler, 1571~1630) inheritedCopernicusThe heliocentric theory reveals the fact that planets orbit the sun in elliptical orbits;Second, Italian physicist Galileo（Galileo, 1564~1642) get the objectOblique projectile motionThe track of isparabola。
It is found that conic curve is not only a static curve attached to the conic surface, but also a universal form of movement of objects in nature.As a result, the processing method of conic curve began to have some small changes.For example, Guidobaldo del Monte (1545~1607) ellipse in 1579 is defined as the trajectory of a moving point with a fixed length from the sum of the distances to two focal points.Thus, the definition of conic curve in the past has been changed.However, the research on the properties of conic curve has not advanced much, and no more new theorems or new proof methods have been proposed.
At the beginning of the 17th century, under the influence of the new idea that a mathematical object can continuously change from one shape to another, Kepler made a new exposition of the nature of conic curve.He found the focus and eccentricity of the conic curve, and pointed out that the parabola has a focus at infinity, and the straight line is a circle passing through infinity.Thus, he was the first to grasp the fact that ellipse, parabola, hyperbola, circle and degenerate conic curve composed of two straight lines can be changed from one to the other continuously, and only various moving modes of focus need to be considered, which provides a logical and intuitive basis for the modern unified definition of conic curve.
along withProjective geometryThe original method of projection and truncation that helped painters may be due to its relationship withconeIt has natural connection and is also used in the research of conic curve.Three French mathematicians in this fieldDeshage（Desargue1591－1661）、Pascal(Pascal, 1623-1662) and Phailippe de La Hire (1640-1718) obtained some special theorems on conic curves, which can be called a new start.But when two other French mathematicians, Descartes and Fermat, created analytic geometry, people's understanding of conic curve entered a new stageresearch methodIs different fromApolloni, which is different fromprojectAnd truncation, but towardsanalytic methodThe direction of development, that is, through the establishment of a coordinate systemequationAnd then use the equation to study the conic curve, in order to get rid of the geometric intuition and achieve the objective of abstraction, and also to obtain the generalization and unity of the conic curve research height.
By the 18th century, people had extensively discussed analytic geometry, exceptRectangular coordinate systemAnd establishPolar coordinate systemAnd can convert these two coordinate systems to each other.In this case, theQuadratic equationIt has also been transformed into several standard forms or introduced curveParametric equation。In 1745, Euler published Introduction to Analysis, which is an important work in the history of analytic geometry and a classic work of conic research.In this book, Euler gave a systematic exposition of conic curve in modern form. Starting from the general quadratic equation, various situations of conic curvecoordinate transformation , one of the following standard forms can always be adopted:EulerLater, 3D analytic geometry also flourished, and many important surfaces were derived from conic curves, such as cylinder, ellipsoid, univalent and bivalveHyperboloidAnd variousparaboloidEtc.
In a word, conic curve plays an important role in mathematics and other scientific and technological fields, as well as in our real life. People have deepened their research on it, and its research results have been widely applied.This just reflects the purpose and law of people's understanding of things.
Related works in the 19th century[7]
Here, we would like to mention the new achievements of our mathematics teacher Hu Xinping in 2016. For thousands of years,Plane analytic geometryThere has been no major progress in the main theory of the focus guide line system. Even though it has been 1700 years since Pappus first discovered the unity of the focus guide line system over 300 A.D., its obvious shortcomings have not been improved.In fact, people have also been seeking to unify seven types of quadratic curves in a geometric way, and Mr. Hu Xinping has given a geometric unified form that includes all eight types of curves of the first and second degree curves. This unified form is the generalization of the unity under the focus directrix, and is also the only complete unity of what form has been seen, which makes the plane analytic geometry take a milestone step forward.It is also a deep mark left by Chinese mathematicians in the development history of plane analytic geometry.
The conic curve usually mentioned includes ellipse, hyperbola and parabola, but strictly speaking, it also includes some degenerate cases.Specifically:
1) When the plane andSecondary coneThe generatrix of is parallel, but not the vertex of the cone. The result is a parabola.
2) When the plane andSecondary coneThe generatrix of is parallel and passes through the vertex of the cone, resulting in a straight line.
3) When the plane is only connected withSecondary coneOne side intersects, but not the vertex of the cone. The result is an ellipse.
4) When the plane is only connected withSecondary coneOne side intersects, but not the vertex of the cone, and is perpendicular to the axis of symmetry of the cone. The result is a circle.
5) When the plane andSecondary coneBoth sides intersect, but not the vertex of the cone, and the result is a hyperbola (each branch isSecondary coneThe intersection line of a cone face and a plane in).
6) When the plane andSecondary coneBoth sides intersect and cross the cone vertex, resulting in two intersecting lines.
7) When the plane does not intersect both sides of the conic surface and passes through the cone vertex, the result is a point.
Note that there is no conic curve in the above curve class: two parallel straight lines.
Algebraic viewpoint
In Cartesian plane, binary quadratic equation
The image of is called a conic.according toDiscriminantIt includes ellipse, hyperbola, parabola and various degenerate cases.
Focus - guide line and its promotion view
（1） Traditional focus guide line unified definition
Given a point P, a straight line l and a constant e>0, then the locus of the point whose distance to P is e in ratio to the distance to l is a conic curve.
The curve varies according to the range of e.The details are as follows:
① E=1 (that is, the distance from P to l is the same), and the trajectory is a parabola;
② 0<e<1, the trajectory is an ellipse;
③ E>1, the trajectory is hyperbolic.
This definition is only applicable to the main cases of conic curve (ellipse, hyperbola, parabola), so it cannot be regarded as a complete definition of conic curve.But because of its simple and beautiful form, and can lead to many important geometric concepts and properties of conic, it is favored and widely used.
（2） Unified definition of primary and secondary curves
（《Mathematical bulletin》In the article "Unification and Nature of the Tracks of Primary and Secondary Curves" in 2016.12, Hu Xinping, a middle school mathematics teacher in China, popularized the focus -- directrix, thus giving the following complete unified definitions of primary and secondary curves)
There are two lines l, m on the plane that are perpendicular to each other and intersect at point E. Point F is a certain point on line m, | EF |=p, point N (called parameter point) is a moving point on line l, and track moving point M meets the following two conditions at the same time:
Picture 3
(I) The directional distance from moving point N and moving point M to fixed line m Nm and Mm have
Nm=(1+t) Mm, where t is a real constant;
(II) Distance from moving point M to fixed point F | MF | Distance from moving point N | MN | Yes
|MF |=e | MN |, where e is a non negative constant,
From the perspective of rectangular coordinate transformation, the locus of moving point M is a conic or a conic
(The agreement e=1, | t |=1, and p=0 are different).
The track of point M is as follows:
(A) When p ≠ 0: there are six classes of conic and conic curves.
When e ≠ 0,
(1) When e=1, | t |=1, the trajectory is a straight line (EFVertical bisector）；
Figure 1
(2) When e=1, | t | ≠ 1, the trajectory is a parabola;
(3) When e<1, e | t |<1, or e>1, e | t |>1, the trajectory is an ellipse, where | t |=1 is a circle;
(4) When e ≠ 1, e | t |=1, the trajectory is two parallel lines;
(5) When e<1, e | t |>1, or e>1, e | t |<1, the trajectory is hyperbolic;
When e=0, the track is a point
(B) When p=0, there are three types of primary and secondary curves.
(1) When e<1, e | t |>1, or e>1, e | t |<1, the trajectory is two intersecting lines;
(2) When e=1, e | t | ≠ 1, or e ≠ 1, e | t |=1, the trajectory is two coincident lines;
(3) When e<1, e | t |<1, or e>1, e | t |>1, the trajectory is a point
The fixed point F and the fixed line l are called the quasi focus of the corresponding trajectory curve and the quasi focus corresponding to the quasi focus FGuide line。
Projective viewpoint
In the projective plane, the locus of the intersection of two projective harnesses with different centers is a conic curve.Two projective points with different bottomsenvelopeIt is a conic curve.
The so-called projective harness means that given two centers O and O ', four lines a, b, c, d and a', b ', c', d 'are drawn from O and O' respectivelyCross ratioCorrespondence is equal, that is, (ab, cd)=(a'b ', c'd'), then these two harnesses are called projective harnesses.The intersection of the corresponding lines of the projective harness (a and a ', b and b', c and c ', d and d' in the above example) must be located on a conic curve.
Similarly, the so-called projective point column refers to that, given two bottom o and o ', four points A, B, C, D and A', B ', C', D 'are taken on o and o' respectively. If the cross ratios of the four points are equal, that is, (AB, CD)=(A 'B', C 'D'), then the two point columns are called projective point columns.The line connecting the corresponding points of the projective point column (A and A ', B and B', C and C ', D and D' in the above example) must be tangent to a conic curve.
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(The following describes the general concepts and properties of the main conic curve in a purely geometric way. Since most properties are defined in the viewpoint of focus guide line, some concepts may not be applicable to more general degradation situations.)
Consider the definition of conic curve from the viewpoint of focus guide line.
focus
The fixed point mentioned in the definition is called thefocus。
Guide line
The fixed line mentioned in the definition is called conicGuide line。
Eccentricity
Fixed constant (i.e. from a point on the conic curve tofocusCorresponding toGuide lineIs called conicEccentricity。
Focal length
The distance from the focus to the corresponding guide line is calledFocal length。
Similar to a circle, the line segment between any two points on a conic is calledstring；The chord passing through the focus is calledFocus chord。The focus chord parallel to the guide line is calledDrift diameter, also known asSine chord。
The conic curve is smooth, so there are tangents andnormalThe concept of.
For the same ellipse or hyperbola, it can be obtained by the combination of two "focus directrix".Therefore, the ellipse and hyperbola have two focal points and two directrix lines.The parabola has only one focus and one guide line.
The conic curve is an axisymmetric figure, and the axis of symmetry is a straight line passing through the focus and perpendicular to the guide line.In the case of ellipse and hyperbola, the line passes through two focal points, which is called theFocal axis。For ellipses and hyperbolasVertical bisectorSymmetry, so ellipse and hyperbola have two axes of symmetry.
Pappus theorem: The focal radius length of a point on the conic curve is equal to the distance from the point to the corresponding guide line multiplied byEccentricity。
Pascal theorem: An inscribed hexagon of a conic curve. If the opposite sides are not parallel, the intersection points of the extension lines on the opposite sides of the hexagon are collinear.(also applicable to degradation)
Brianchon theorem: conicCircumscribeA hexagon whose three diagonals are at the same point.
Ice cream theorem (consistency of definitions)
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By Belgian mathematician GF. Dandelin, 1822Ice cream theoremIt is proved that the geometric definition of conic curve is equivalent to the definition of focus guide line.
That is, there is a cone (eggshell) with Q as the vertex, and a plane π '(you can also say biscuit) intersects with it to obtain a conic curve. The sphere is tangent to the plane π' and the cone. When the curve is an ellipse or hyperbola, the plane and the sphere have two tangent points. If there is only one parabola (or the other is at infinity), the tangent point is the focus.The intersection of the sphere and the cone is a circle. If the intersection of the plane π and π 'where the circle is located is a straight line d (when the curve is a circle, d is an infinite line), then d is the guide line.
Conic curve
The figure only draws an ellipse, proving that it is applicable to parabola hyperbola, that is, any tangent point is the focus, and d is the directrix.
Certificate: Assume that P is a point on the curve, and the line PQ intersects the circle O at E.Let the intersection angle of plane π′ and π be α, and the generatrix of cone (such as PQ) and plane πAngle of intersectionIs β.Let the perpendicular of P to plane π be H, and the perpendicular of H to line d be R, then PR is P to dvertical（Three perpendicular theorem）, and ∠ PRH=α.Because PE and PF are both sphericaltangent, PE=PF.
The properties of ellipse, hyperbola and parabola can refer to the corresponding terms, and the properties of general conic curve are given.
Theorem 1: If any three of the five points in the plane are not collinear, there is only one conic curve passing through these five points.
Theorem 1: If any three of the five straight lines in the plane do not share a common point, then there is only one conic tangent to all the five straight lines.
Theorem 2（Pascal's theorem）: The intersection points of three groups of opposite sides of a hexagon inscribed on a nondegenerate conic curve (ellipse, hyperbola, parabola, circle) are collinear.
Theorem 2‘（Brion's theorem）: The three diagonals of a hexagon circumscribed by a nondegenerate conic curve (ellipse, hyperbola, parabola, circle) share a common point.
Theorem 3 (the inverse of theorem 2): If the intersection points of three opposite sides of a hexagon are collinear, then the hexagon is inscribed on a conic curve.
Theorem 3 (the inverse of Theorem 2): If three diagonals of a hexagon share a common point, then the hexagon is circumscribed on a conic curve.
Theorem 4 (the limit case of Theorem 2): the inscribed triangle of a conic curve, the tangent of each vertex is collinear with the intersection point of its opposite side.Let △ ABC be inscribed on a conic curve, the tangent at point A is a, the tangent at point B is b, and the tangent at point C is c.If a and BC intersect with P, b and AC intersect with Q, and c and AB intersect with R, then the PQR three points are collinear.
Theorem 4 (the limit case of Theorem 2): The tangent point of each side of the circumscribed triangle of a conic curve shares a common point with the line connecting it to the vertex.Let △ ABC circumscribe a conic curve, the tangent point on BC side is P, the tangent point on AC side is Q, and the tangent point on AB side is R.Connect AP, BQ and CR, then the three straight lines intersect at the same point.
1、 For the unified equations and properties of quadratic curves, please refer to the article "Unification and Properties of the Tracks of Primary and Secondary Curves" in Mathematics Bulletin 2016, Issue 12.
① When e=1, it means that F (g, h) is the focus, and p is the focus toGuide lineDistancedparabola。among
The included angle α with the polar axis (A is the vertex of the parabola).
② 01 (g, h) is a focus, and p is the focus toGuide lineDistance, e is the eccentricityellipse。among
Angle with polar axis α.
③ When e>1, it means Ftwo(g, h) is a focus, and p is the focus toGuide lineDistance, e is the eccentricityhyperbola。among
Angle with polar axis α.
④ When e=0, it indicates point F (g, h).
This unified equation can be used to solve the conic curve in the plane by the five point method.Substitute five groups of ordered real number pairs to find the corresponding parameters.
Note: This equation is not applicable to other degenerate forms of conic, such ascircularEtc.
Appendix: When e ≠ 0, F (g, h) corresponds to the directrix equation:
Discriminant method
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In the plane rectangular coordinate system, the equation
The curve represented by is called a conic curve, where a, b, c are different from zero at the same time, and it includes the following nine types of curves:
Conic curve
Recorded in the table
Ione,I2,IthreeAnd JtwoThey are called the invariants of the conic (that is, they are invariant after coordinate transformation) and the conditional invariants (or semi invariants).[4]
Conical curve and straight line slope
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In college entrance examination, we often encounter problems related to the fixed value of the sum, product and quotient of the slopes of two moving straight lines.Now, we will focus on the conic curve (ellipse, hyperbola, parabola) on the x-axis to study the general practice of such problems and the underlying truth.
The sum of slopes is a fixed value
If the sum of slopes is a fixed value, it must be equal toHarmonic point seriesRelevant, that is, under this type of topic, it is sure to find a set of harmonic points (harmonic harnesses).In conic curve, only poles are related to harmonic point seriesPolar lineHowever, since the method of pole and polar line cannot be used in the college entrance examination, the principle is explained only through this method, and Weida's theorem or homogeneous simultaneous method is required in the actual question.
(1) The intersection of two moving straight lines is a fixed point on the conic curve
That is, two straight lines AC and AD are drawn from a point on the conic curve. If CD passes through the fixed point B, then kAC+kADIs a fixed value.Conversely, if k is knownAC+kADFor a fixed value, the CD can also be pushed through a fixed point B.
The sum of slopes is a fixed value
As shown in the figure, A is the fixed point on the conic curve, and A 'is the symmetric point of A about the x-axis.Find a point B on the tangent line passing through A ', make a secant CD through B, and connect AC and AD. Then there are two moving lines AC and AD, whose intersection point is the fixed point A on the conic curve and passes through the fixed point B.
Go through B and make another tangent BE of the conic curve. If the tangent point is E, then A'E is the polar line of B.
Connect AB again, intersect the conic curve at another point I, and connect CI (not shown in the figure) to get the inscribed quadrilateral AICD of the conic curve.Let two groups of opposite sides intersect with B and J respectively (J can beInfinity point, i.e. AD ‖ CI), the diagonal intersects K, according to the harmonic property of the inscribed quadrilateral of the conic curve, JK is the polar line of B, and JEKA 'collinear is obtained.
At the same time, BK is also the polar line of J, so according to the definition of the polar line, J, K, E, A 'form the harmonic point column ⇒ AD, AC, AE, AA' form the harmonic harness ⇒ N, M, Q, P form the harmonic point column ⇒
⇒
⇒
。
Because B is a fixed point and BE is a tangent, E is also a fixed point, so kAEIs a fixed value, so kAC+kADIs a fixed value.
Conversely, if k is knownAC+kADIs a fixed value, which can be set to
, the fixed line AE is obtained.Then, make tangent lines through E and A 'respectively, and the tangent lines intersect with B, then from the above derivation, we can know that B is the fixed point through which CD passes.
Two situations should be noted here:
a. For a central conic curve (ellipse, hyperbola), when
It is easy to prove that E and A 'are symmetrical about the center, so the tangent lines at E and A' are parallel.At this time, it is considered that the two tangents intersect at the infinity point B. CD passing through the infinity point means that no matter how CD changes, its slope is a fixed value and equal to the slope of the tangent at E and A '(because the finite lines passing through the same infinity point B are parallel to each other, and the slopes of the parallel lines are equal).
For centerless conic curve (parabola), when
When AE ‖ x axis, AE and x axis intersect at infinity P.According to the theory of projective geometry, P is the tangent point of the parabola and the infinite straight line, that is, AE intersects the parabola at P, and the tangent passing through P is the infinite straight line.When making a tangent after A ', the tangent line and the infinity line also intersect at the infinity point B. Therefore, the slope of CD is still a fixed value and equal to the slope of the tangent at A'.
b. When AE is exactly the tangent at A, there is only one intersection point A between AE and conic curve.It may be considered that A and E coincide, so the tangent passing through E (A) and the tangent passing through A 'intersect on the x axis (that is, the pole of the straight line AA').
(2) The intersection point of two moving straight lines is a fixed point not on the conic curve
Different from the previous case, in this case, a straight line is drawn through the fixed point A not on the conic curve to intersect the conic curve at two points P, Q, and B as the fixed point (not on the conic curve)BP+kBQIt is a fixed value.Or ask whether there is a fixed point B to make kBP+kBQIt is a known fixed value.
Here we take the simplest case to analyze (in fact, this kind of question is also the most popular in the college entrance examination).As shown in the figure, A is a certain point on the x-axis (not on the conic curve), through which a straight line PQ intersects the conic curve at P and Q. B is a certain point on the polar line of A (note that this is a prerequisite, B must be on the polar line of A, otherwise kBP+kBQIs not a fixed value), calculate kBP+kBQ。
The sum of slopes is a fixed value
Let the polar line of PQ intersection A be R, then according to the definition of polar line, P, Q, A, R form a harmonic point series ⇒ S, T, A, U are also harmonic point series ⇒
⇒
⇒
。
Conversely, if k is knownBP+kBQIs a fixed value, which can be set to
, get the fixed line AB, and the polar intersection point of AB and A is the obtained B.
The product of slope is a fixed value
If the slope product is a fixed value, there is no limit, because in theProjective geometryIn, the classification of a conic is determined by the position of a conic and an infinite straight line, that is, by translating an infinite straight line, a conic can become an ellipse (away from an infinite straight line), a parabola (tangent to an infinite straight line), or a hyperbola (intersecting an infinite straight line).Since the translation is an infinite straight line, the conic itself does not change, so we can only use the ellipse to explore the truth behind it.
A circle is a special ellipse, that is, a circle can become an ellipse through affine transformation.
(1) The intersection of two moving lines is a fixed point on the circle
The product of slope is a fixed value
As shown in the figure, AB is a diameter of a circle, and two chords AP and AQ are led out from point A on the circle. If PQ passes point M on AB, then kAPkAQIs a fixed value.Conversely, if kAPkAQIs a fixed value, then PQ passes through the fixed point M.
Using the triangle area formula,
。
When AB is the x-axis, tan α tan β is just kAPkAQ。So when M is a fixed point, the left ratio is a fixed value, so the right kAPkAQIs a fixed value.Conversely, when kAPkAQWhen it is a fixed value, the ratio on the left is also a fixed value, soCoordinate formula of fixed proportion dividing pointIt can be seen that M is the fixed point.
(2) The intersection of two moving lines is a fixed point not on the circle
Different from the previous case, in this case, a straight line is drawn through the fixed point M not on the circle to intersect the conic curve at two points P and Q, and N is the fixed point (not on the conic curve).Connect PN and QN to intersect with S and T, then the straight line ST passes through the fixed point.
At first glance, it seems to have nothing to do with the slope product, but in fact, it is not.The first case is needed to prove this conclusion.
Conic curve
As shown in the figure, AB is the diameter of the circle, M and N are on AB, and the intersection points have been marked in the figure.
In order to prove ST over point (set to K), it is only necessary to prove
Is a fixed value.
and
According to the conclusion of the first case, the straight line PS and QT pass through point N, while the PQ pass through point M ⇒ the numerator and denominator are both fixed values ⇒ ST pass through point K.
The reverse is not true, that is, it is impossible to push PS and QT from PQ to point M and ST to point N.This is because although the denominator and fractional values in the above formula are fixed values, the numerator is also a fixed value.However, the molecule is the product of two numbers, and each factor can change as long as the multiplication remains unchanged, so PS and QT do not necessarily pass through the fixed point.
Note: The above fixed-point M and N can not be in the circle, but can be at any position on the straight line AB (except A and B). It is proved that they are similar and the conclusion is the same.
In addition, when the circle is stretched to an ellipse, for example, the ordinate of the circle is changed to the original t (t ≠ 0) times, becauseSingle ratioIs an affine transformation invariant, so BM/AM in (1) is invariant.And kAP'=tkAP，kAQ‘=tkAQ, so there is
unchanged.
According to the third definition of ellipse,
Is a fixed value.If M is a fixed point, then kAPkAQIs a fixed value, so kBPkAQIs a fixed value.Conversely, if kBPkAQIt is a fixed value, and there is also PQ over point M.
The slope ratio is a fixed value
This is an important corollary of the case (2) where the slope product is a fixed value, that is, there is
Is a fixed value.
The proof is very simple. Let the straight line PQ and ST intersect at point R, pass through R and make RH ∨ AB, and the vertical foot is H.
According to the nature of conic inscribed quadrilateral, R is on the polar line l.And because N is on AB, there is l ∨ AB, so RH is just the polar line of N.Because no matter how R changes, the vertical H is constant.And kPQ=HR/MH，kST=HR/KH, so there are
。
K. M and H are fixed points, so the ratio is a fixed value.
Theorem Introduction
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CGY-EH theorem (also calledHard solution theorem of conic curve）It is a set of solutions for ∆, x1+x2, x1 * x2, y1+y2, y1 * y2 and intersecting chord length when ellipse/hyperbola intersects with straight linesimple and convenientAlgorithm
Theorem content
If the curve
Intersect with straight line A χ+By+C=0 at two points E and F, then:
To solve y1+y2 and y1 * y2, simply exchange the values of A and B, and the values of m and n. It can be seen that the values of ε and ∆ 'will not change accordingly.
Theorem supplement
Simultaneous curve equation and y=kx+
It is a more common phenomenon in the current college entrance examination than the simultaneous "Ax+By+C=0". The quadratic equation after simultaneous establishment is an indispensable item in the standard answer. x1+x2, x1x2 can be directly obtained through this equation and Veda's theorem, but the expression of chord length needs a lot of calculation.Here is a simplified slope formula of CGY-EH to reduce the amount of memory for application in the exam.
If the curve
And straight line y=kx+
Intersect at E and F, then:
there
It can be either a constant or an algebraic expression about k.From this formula, we can deduce:
If the curve
Is an ellipse
, then
If the curve
Hyperbolic
, then
Since the CGY-EH theorem can not be directly applied in the college entrance examination, students can only get full step points by answering this question (the content of ellipsis needs to be filled in by the examinee):
... (quadratic formula) of simultaneous two equations (*)
So x1+x2=...... ①, x1x2=...... ②;
So | x1-x2 |=√ (x1+x2) ^ 2-4x1x2=...... (At this time, substitute formulas ① and ② to get a large formula, but do not simplify)
Reduced | x1-x2|=
(Formulas are directly set up secretly without simplification)
Then we can find the chord length
Has.
Simple proof of theorem
Let the curve x ^ 2/m+y ^ 2/n=1 ① intersect with the straight line A χ+By+C=0 ② at two points E and F, and the final quadratic equation can be obtained by simultaneous formula ① ②:
aboutequivalenceThe value of the unary quadratic equation ∆ of Δ is not unique, and the meaning of ∆ is only its relationship with zero, so it is always true by 4B ^ 2>0, then ∆ '=mn (ε - C ^ 2) with the same sign as ∆ can be taken as the value of ∆.
By | EF |=√ ([(x_1-x_2)] ^ 2+[(y_1-y_2)] ^ 2)=√ ((1+A ^ 2/B ^ 2) [[(x_1+x_2)] ^ 2-4x_1 x_2])
Available | EF |=√ ((A ^ 2+B ^ 2) 4mn (A ^ 2 m+B ^ 2 n-C ^ 2))/(| A ^ 2 m+B ^ 2 n |)
The conic curve includes ellipse, parabola, hyperbola and circle. Through the rectangular coordinate system, they correspond to the quadratic equation, so the conic curve is also calledConic。Conic curve has always been one of the important subjects in geometry research, and there are many conic curves in our real life.
The earth we live on runs on an elliptical orbit around the sun every moment, as do other planets in the solar system. The sun is located in an elliptical orbitfocusOn.If the speed of these planets increases to a certain extent, they will travel along a parabola or hyperbola.Human launchArtificial earth satelliteOr artificial planets should follow this principle.With respect to an object, pressLaw of universal gravitationThe motion of another object attracted by it cannot have any other orbit.Therefore, in this sense, conic curve constitutes the basic form of our universe.
By rotating the parabola around its axis, we can get a surface called the rotating object surface.It also has an axis, the axis of the parabola.There is a focus with wonderful properties on this axis, and any straight line passing through the focus isparaboloidAfter reflection, it becomes a straight line parallel to the axis.This is why we should make the searchlight reflector into a rotating paraboloid.By rotating the hyperbola around its imaginary axis, we can getHyperboloid of one sheetIt is also a ruled surface, which is composed of two groups of parent lines. The parent lines in each group do not intersect each other, but intersect with another group of parent lines.People are designing tall towers (such asCooling tower）The shape of hyperboloid of one sheet is adopted, which is light and strong.
It can be seen that the value of conic curve cannot be overestimated in any case.
optical properties
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ellipse
The light emitted from one focus of the ellipse, after being reflected by the ellipse,Reflected lightAre converging into another ellipsefocusOn.
hyperbola
The light emitted from one focus of the hyperbola passes through the hyperbolareflexAfter the reflection of lightReverse extenderThey all converge on the other focus of the hyperbola.
parabola
The light emitted from the focus of the parabola is parallel to the axis of symmetry of the parabola after being reflected by the parabola.
a tuft ofDirectional lightThe collimation line perpendicular to the parabola shoots into the opening of the parabola. After being reflected by the parabola, the reflected light will converge at the focus of the parabola.
application
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Acoustic Properties of Ellipse
As shown in the figure, echo valley is the application of ellipse in conic curve.Some ellipses in the westdomeThis phenomenon also exists in the cathedral of.
Conical section is important in astronomy: the orbits of two huge objects interacting according to Newton's law of universal gravitation are conical sections if their common center of mass is considered static.If they are bound together, they will track the ellipse;If they separate, they will follow a parabola or hyperbola.See the two body problem.
For some fossils in paleontology, knowing the cone section can help to understand the three-dimensional shape of some organisms.
The reflective properties of the conical section are used for searchlights,radio telescopeAnd someOptical telescopeDesign of[5]。useParabolic mirrorAs a reflector, use the bulb at the focus under the searchlight.The 4.2-meter Herschel Optical Telescope in La Palma, Canary Islands, uses the primary parabolic mirror to reflect light to the secondary hyperbolic mirror, which reflects that it once again becomes the focus behind the first mirror.
