parabola

One of conic curves

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Parabola refers to the locus of points in the plane with the same distance from a certain point and a certain straight line (the fixed line does not pass through the fixed point), where the fixed point is called the focus of the parabola and the fixed line is called the parabola Guide line 。 It has many representations, such as parameter representation, standard equation representation, and so on. It has important applications in geometrical optics and mechanics. Parabola is also a kind of conic curve, that is, the cone surface is parallel to a a bus or bus bar The plane of curve 。 Parabola in the appropriate coordinate transformation Next, it can also be regarded as Quadratic function image 。^[3]

Chinese name: parabola
Foreign name: Parabola
Alias: Conic parabola
expression: y=ax^2+bx+c

Presenter: Apollonius （Apollonius）
Proposed time: ancient Greek times
Applicable fields: analytic geometry Dynamics optics
Applied discipline: mathematics Physics
Classification: A type of conic curve

catalog

▪ Geometric properties of tangents and normals
▪ Geometric properties of strings
▪ prove
seven Drawing with ruler and gauge of tangent line
eight Analytic solution

brief introduction

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In mathematics, a parabola is a Plane curve , it is mirror symmetric, and when the orientation is roughly U-shaped (if different directions, it is still a parabola). It is applicable to any one of several seemingly different mathematical descriptions, which can be proved to be identical curves.

A description of a parabola involves a point (focal point) and a line (guide line). The focus is not on the guide line. The parabola is the locus of a point in the plane equidistant from the directrix and focus. Another description of a parabola is that it is a conical section formed by the intersection of a conical surface and a plane parallel to the conical generatrix. The third description is algebra.

The line perpendicular to the directrix and passing through the focus (that is, the line passing through the middle decomposition parabola) is called the "axis of symmetry". The point on the parabola that intersects the axis of symmetry is called the "vertex", and it is the sharpest curved point of the parabola. The distance between the vertex and the focus measured along the axis of symmetry is the "focal length". A "straight line" is a parallel line of a parabola and passes through the focal point. The parabola can be opened up, down, left, right or in any other direction. Any parabola can be repositioned and repositioned to fit any other parabola - that is, all parabolas are geometrically similar.

Parabola has such a property that if they are made of reflective materials, the light traveling parallel to the symmetric axis of the parabola and hitting its concave surface will be reflected to its focus, regardless of where the parabola is reflected. Instead, the light generated from the point source at the focus is reflected into a parallel ("collimated") beam, making the parabola parallel to the axis of symmetry. Sound and other forms of energy have the same effect. This reflective property is the basis of many practical applications of parabola.

Parabola has many important applications, from Parabolic antenna Or parabolic microphone to vehicle headlamp reflector to design ballistic missile 。 They are often used in physics, engineering and many other fields.

development history

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Apollonius Eight volumes《 Conic curve 》(Conics) ancient Greek Analytic geometry One reach the peak of perfection It is the work of fine craftsmanship. The ellipse we all know today（ ellipse ）Parabola, Hyperbola（ hyperbola ）These nouns were invented by Apollonius. At that time, the study of this simple and perfect curve was purely based on geometry To study the curve closely related to the circle; Their geometry is a natural generalization of the circular geometry, which was a pure exploration of ideas in those days, and we can neither hope nor expect that they will really play an important role in the basic structure of nature.

Parabolic problem

Standard equation

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definition

Parabola of right opening:

Left opening parabola:

Upper opening parabola:

Lower opening parabola:

[p is Focal length ]

characteristic

In parabola

Medium, the focus is

, alignment equation yes

, eccentricity

, Range:

；

In parabola

Medium, the focus is

, alignment equation yes

, eccentricity

, Range:

；

In parabola

Medium, the focus is

, the equation of the guide line is

, eccentricity

, Range:

；

parabola

In parabola

Medium, the focus is

, the equation of the guide line is

, eccentricity

, Range:

。

Four equations

Similarities and Differences of Four Parabolic Equations

common ground:

① origin On the parabola, the eccentricity e is 1 ② Axis of symmetry by Axis ；

③ Guide line Perpendicular to the axis of symmetry, the vertical foot and the focus are symmetrical to the origin respectively, and their distances from the origin are equal to Coefficient of primary term Of absolute value 1/4 of

difference:

① When the symmetry axis is the x axis, the right end of the equation is ± 2px, and the left end of the equation is y ^ 2; When the symmetry axis is y-axis, the right end of the equation is ± 2py, and the left end of the equation is x ^ 2;

② Opening direction and x (or y) axis Positive halfshaft In the same case, the focus is on the positive half axis of the x-axis (y-axis), and the right end of the equation is taken as Positive sign ； Opening direction and x (or y axis) Negative halfshaft In the same case, the focus is on the negative half axis of the x-axis (or y-axis), and the right end of the equation is taken as minus sign 。

Tangent equation

Parabola y two =2px Previous point (x zero ，y zero ）The tangent equation at is:

。

Parabola y two =The equation of the slope of the passing focus on 2px is: y=k (x-p/2).

Related parameters

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(For parabola y with right opening two =2px）

The image of quadratic function is a parabola

Eccentricity : e=1 (constant value, which is the distance between a point on the parabola and the guide line and the distance ratio between the point and the focus)

focus :（p/2，0）

Directrix equation l:x=-p/2

vertex ：（0，0）

Define Fields : for parabola y two =2px, when p>0, the definition field is x ≥ 0, when p<0, the definition field is x ≤ 0; For parabola x two =2py, the definition field is R.

range : for parabola y two =2px, range R, for parabolic x two =2py, when p>0, the value range is y ≥ 0; when p<0, the value range is y ≤ 0.

Explanation of terms

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Guide line Focus: A parabola is the locus of a point in the plane that is equidistant from a certain point and a fixed line that is not at this point. This fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola.

Axis: parabola is axial symmetry Graph, its Axis of symmetry It is called shaft for short.

Focus distance: The distance from the focus to the guide line is called the focus distance, and the length is p.

Focal radius : The line segment obtained by connecting any point on the parabola with the focus of the parabola. For parabola y two =2px，P(x zero ，y zero ）, then | PF |=x zero +p/2。

string : The chord of a parabola is a line segment connecting any two points on the parabola.

Focal chord : The focal chord of the parabola is the chord passing through the focus of the parabola. For parabola y two =2px，A(x one ，y one )，B(x two ，y two ）, then | AB |=x one +x two +p=2p/sin two θ （ θ Is the inclination angle of AB)

Sine chord : The positive focal chord of the parabola is the focal chord perpendicular to the axis, also called path. The drift diameter is 2p.

diameter : The diameter of a parabola is the locus of the midpoint of a group of parallel chords of the parabola. This diameter is also called the Conjugate diameter 。 All diameters are parallel to the axis, so the diameter of a parabola can also be defined as a parallel line (ray) passing through any point on the parabola as the axis

Main diameter: The main diameter of the parabola is part of the axis of the parabola (rays inside the parabola).

Parabola refers to the curve that an object passes through in the air when it is thrown and falls on the distant ground.

Geometric properties

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Geometric properties of tangents and normals

(1) Let the tangent line and directrix of a point P on the parabola intersect at Q, and F is the focus of the parabola, then PF ∨ QF. If PA is perpendicular to the alignment through P, and the perpendicular foot is A, then PQ bisects ∠ APF.

(2) Pass a point P on the parabola as the vertical line PA of the guide line, then the bisector of ∠ APF is tangent to the parabola at P Is the property (1) the inverse theorem of the second part) Drawing with ruler and gauge method.

(3) Let the tangent of a point P (P is not a vertex) on the parabola be normal If the axes are respectively at A and B, then F is the midpoint of AB. This property can deduce the optical property of the parabola, that is light Meridional parabola reflex The rear ray is parallel to the axis of symmetry of the parabola. various searchlight The automobile lamp uses the nature of parabola (surface) to make light source Emit at the focus (accurate) Directional light 。

(4) Let the tangent axis of point P on the parabola, except the vertex, be A, and the tangent of intersection vertex O be B, then FB is vertically bisected PA, and the intersection point M of FB and the directrix is exactly the projection of P on the directrix (that is, PM is perpendicular to the directrix).

(5) A triangle formed by three tangent lines of a parabola, whose circumscribed circle passes through the focus. That is, if AB, AC and BC are tangent lines of parabola, the four points of ABCF are in the same circle.

(6) Pass a point P outside the parabola to make two tangent lines of the parabola, and the chord connecting the tangent point intersects with the axis at A. If the projection of P on the axis is B, then O is the midpoint of AB.

(7) If the parabola is tangent to all three sides (the straight line) of a triangle, the guide line passes through the perpendicular center of the triangle.

Geometric properties of strings

(8) The tangents at both ends of the focus chord are perpendicular to each other and perpendicular to the guide line.

(9) The endpoints A and B of the focus chord are used as the perpendicular lines of the guide line, and the perpendicular feet are M and N respectively. Let the tangents at A and B intersect at P, then P is the midpoint of MN, and the tangent line of the circle with diameter AB is at P.

(10) If the two focus chords of the parabola are equal, and the midpoint of the two focus chords is connected, the line is perpendicular to the axis.

(11) A chord AB of the parabola intersects with the axis at P (not necessarily the focus F), passes through A and B as perpendicular lines AM and BN of the axis respectively, and the apex of the parabola is O, then OP ²= AM*BN。

prove

The above properties can be proved by coordinate method

Give examples to prove the properties (1), (3), (4) and (9).

(1) Focus

, Alignment

, set

, then the tangent equation passing through P is:

order

, get

, so

therefore

，

It is easy to prove that the quantity product of the two is 0, so there is PF ∨ QF.

To prove that PQ bisects ∠ APF, we can prove that Rt △ APQ ∨ Rt △ FPQ through the judgement method HL of congruent triangles, and then we can get the corresponding angle ∠ APQ=∠ FPQ. HL is obvious, because according to the definition of parabola, there is PF=PA, and the hypotenuse PQ is a common edge, so the two triangles are congruent.

According to this property, we can also draw a corollary: AF is vertically bisected by PQ, and the quadrilateral PAQF is inscribed on the circle, and PQ is the diameter.

(3) Let a point on the parabola except the vertex

, then the equation of tangent l passing P is:

Organize it into

According to the linear Point normal equation As you can see, a normal vector of l is

, and passes through the point

, i.e

。

For the normal m at point P, the normal vector of l

Is the direction vector of m. According to the pointwise equation of the straight line, the equation of m is

。 Let y=0, the solution is x=x zero +p. I.e

。

According to the midpoint coordinate formula,

Is the midpoint of AB.

As the PQ ‖ x axis, to prove the optical properties of the parabola Reflection law of light It can be seen that as long as it is proved that the reflection angle ∠ QPB is equal to the incident angle ∠ FPB, that is, PB bisects ∠ FPQ.

Using plane geometry, according to Middle line theorem of hypotenuse of right triangle It can be seen that PF=BF

∠ FPB=∠ FBP (equal sides and equal angles)

∠ PFA=∠ FPB+∠ FBP=2 ∠ FBP (External angle theorem)

∵PQ∥AB

∠ PFA=∠ FPQ, ∠ FBP=∠ BPQ (two straight lines are parallel, and the internal angle is equal)

∴∠FPQ=2∠BPQ

 PB bisects ∠ FPQ, and the proposition is proved.

(4) According to known conditions, A is on the x axis and B is on the y axis.

The PA equation is:

, let x and y equal 0, the solution is

It is easy to verify that B is the midpoint of AP

and

, their quantity product is 0, so BF ∨ AP, that is, BF vertically bisects AP.

To verify that PM is perpendicular to the alignment, as long as the ordinate of M is the same as P, it is y zero OK.

Easy to write straight line BF:

, Order

, solved

The proposition is proved.

(9) Set

Unilever AB and Parabolic equation , eliminate x

from Veda's theorem ，

PA and PB are tangent lines, according to Tangent equation ，

The expression of simultaneous PA and PB can be solved

and

According to the midpoint coordinate formula and Veda's theorem, P is the midpoint of MN.

If the midpoint of AB is E, then the ordinate of E

, the same as the ordinate of P,

So PE ‖ x axis, PE ∨ MN

According to the property (8), it can be known that PA ∨ PB, that is, △ PAB is a right triangle

So E is the outer center of △ PAB, so PE is the radius

According to the judgment theorem of tangent, MN is the tangent of ⊙ E, and the tangent point is P.

Drawing with ruler and gauge of tangent line

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According to the geometric property (2), we can get the method of drawing with ruler and gauge through a point on the parabola or a point P outside the parabola as the tangent line of the parabola.

(1) P on parabola

① Perpendicular line passing P as the reference line, set A as the vertical foot

② Connect PF (F is the focus)

③ As the bisector PQ of ∠ APF

According to property (2), the line PQ is tangent

(2) P is outside the parabola

① Connect PF

② Draw an arc with P as the center and PF as the radius. The arc and the guide line intersect at A and B respectively

③ Perpendicular lines passing through A and B shall be drawn respectively, and the vertical lines and parabolas shall be intersected at M and N respectively

④ Connect PM and PN, then PM and PN are tangent lines (there are two)

This is because if MF is connected, then in △ PAM and △ PFM

 PA=PF (definition of circle), PM=PM (common edge), MA=MF (definition of parabola)

∴△PAM≌△PFM（SSS）

∠ AMP=∠ FMP (corresponding angles of congruent triangles are equal)

 MP bisector ∠ AMF (definition of angle bisector)

 MP is tangent (property (2))

Similarly, NP can be proved to be another tangent

Analytic solution

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Take the focus on the X axis as an example

Know P (x zero ，y zero ）

Let the demand be y one =2px

Y zero one =2px zero

So 2p=y zero one /x zero

So the parabola is y one =（y zero one /x zero ）x

The summary is as follows:

(1) Know that the parabola passes through three points (x1, y1) (x2, y2) (x3, y3) Let the parabola equation be y=ax ²+ bx+c，

Substitute the coordinates of each point to get a ternary linear equation system, and solve the values of a, b, c to get the analytical formula.

(2) Know the two intersection points (x1, 0), (x2, 0) of the parabola and the x-axis, and know that the parabola passes a certain point (m, n),

Let the equation of the parabola be y=a (x-x1) (x-x2), and then substitute the point (m, n) to obtain the coefficient of the quadratic term a.

(3) Know the symmetry axis x=k,

Let the parabolic equation be y=a (x-k) ²+ b. Then determine the value of a and c in combination with other conditions.

(4) Know that the maximum value of the quadratic function is p,

Let the parabolic equation be y=a (x-k) ²+ p. A, k shall be determined according to other conditions.

Extended Formula

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Parabola: y=ax two + bx + c （a≠0）

Y is equal to the square of ax plus bx plus c;

When a>0, the opening is upward;

When a<0, the opening is downward;

When c=0, the parabola passes through the origin;

When b=0, the symmetric axis of the parabola is y-axis.

also Vertex form y = a（x-h） one + k

H is x of vertex coordinate;

K is y of vertex coordinates;

It is generally used to calculate the maximum and minimum values.

Parabolic standard equation: y one =2px

It means that the focus of the parabola is on the positive half axis of x, and the focus coordinate is (p/2,0). The directrix equation is x=- p/2.

Since the focus of the parabola can be on any half axis, there is a common standard equation y one =2px，y one =-2px，x one =2py，x one =-2py。

Quadratic function image

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stay Rectangular coordinate system Make quadratic function y=ax in two +Bx+c, we can see that the quadratic function image without a specific domain is an endless parabola. If the drawing is correct, the quadratic function image will be

Translated.

The quadratic function image is axial symmetry Figure, the axis of symmetry is a straight line

。

Axis of symmetry and Quadratic function image only The intersection point of is the vertex P of the quadratic function image.

In particular, when b=0, the symmetry axis of the quadratic function image is Y-axis (that is, the line x=0) is the abscissa of the vertex (that is, x=?).

a. B Same number, Axis of symmetry On the left side of the y-axis

a. B Different sign, symmetry axis is on the right side of y-axis

The quadratic function image has a vertex P with coordinates P (h, k).

When h=0, P is on the y-axis; When k=0, P is X axis On. It can be expressed as the vertex formula y=a (x-h) one +k（a≠0）

，

。

Coefficient of quadratic term A determines the opening direction and size of the quadratic function image.

When a>0 Time, quadratic function image Up Opening; When a<0 Time, parabola down Opening.

|a| Larger , the opening of the quadratic function image Smaller 。

The linear term coefficient b and the quadratic term coefficient a jointly determine the position of the axis of symmetry.

When a>0, and b with When (i.e. ab > 0), axis of symmetry at Y-axis left ； Because the symmetry axis is on the left, the symmetry axis less than 0, that is - b/2a<0, so b/2a should be greater than 0, so a and b should be the same number.

When a>0, and b different When (i.e. ab < 0), axis of symmetry at Y-axis right 。 Because the symmetry axis is on the right, the symmetry axis should greater than 0, that is, - b/2a>0, so b/2a should be less than 0, so a and b should be signed differently.

Can be simply memorized as Same left but different right , that is, when a and b are the same sign (that is, ab>0), the symmetry axis is on the left of the y-axis; When a and b are signed differently (that is, ab<0), the axis of symmetry is on the right of the y-axis.

In fact, b has its own geometric meaning: the quadratic function image at the intersection of the quadratic function image and the y-axis tangent Of Function analytic expression （ Linear function ）The value of the slope k of. The quadratic function can be Derivation Get.^[1]

Quadratic function image

Relevant conclusions

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A (x1, y1), B (x2, y2), A, B in parabola y one =2px, there are:

① When straight line AB passes the focus, x one x two = p ²/ 4 ， y one y two = -p ²；

(When A, B are in parabola x ²= 2py, there is x one x two = -p ² ， y one y two = p ²/ 4. It can only be established when a straight line passes through the focus)

② Focus chord length: | AB |=x one +x two +P = 2P/[（sin θ） two ]；

③ （1/|FA|）+（1/|FB|）= 2/P； (The length of the long one is P/(1-cos θ）， The shorter one is P/(1+cos θ））

④ If OA is vertical to OB, AB passes through point M (2P, 0);

⑤ Focal radius : | FP |=x+p/2 (the distance from a point P on the parabola to the focus F is equal to the distance from P to the guide line L);

⑥ Chord length formula ：AB=√（1+k one ）*│x one -x two │；

⑦△=b one -4ac；

⑴△=b one -There are two 4ac>0 Real root ；

⑵△=b one -4ac=0 has two identical real roots;

⑶△=b one -4ac<0 has no real number.

⑧ The distance from the focus of the parabola to the perpendicular of its tangent is the distance from the focus to the tangent point and the distance to the vertex Proportional middle term ；

⑨ The standard form of parabola is at (x zero ，y zero ）The tangent of the point is: yy zero =p（x+x zero ）

(Note: x in tangent equation of conic curve ²= x*x 0 ， y ² =y*y zero ， x=（x+x zero ）/2 ， y=（y+y zero ）/2 ）^[2]

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