General,hyperbola(Greek: "γπε ρβ ολ joint"[3], which literally means "beyond" or "beyond") is defined as a plane intersection right angleconeA class of two halves of a faceConic curve。
It can also be defined as two fixed points (calledfocus）The distance difference of isconstantOf pointstrajectory。The fixed distance difference isaTwice as much as hereaIs from the center of the hyperbola to the nearest branch of the hyperbolavertexDistance.aAlso called the real half axis of the hyperbola.The focus is on the through axis, and their middle point is calledcore, the center is generally located atoriginAt.
In mathematics, hyperbola (multiple hyperbola or hyperbola) is a smooth curve located in a plane, defined by its geometric characteristics or the equation of its solution combination.The hyperbola has two pieces, called connected components or branches, which are mirror images of each other, similar to two infinite bows.
Hyperbola is one of the three cone sections formed by intersection of the plane and bicone.(The other conic parts are parabolas and ellipses, and the circle is a special case of an ellipse) If the plane intersects the two halves of the bicone, but does not pass through the vertex of the cone, the conic curve is hyperbolic.
Hyperbola appears in many ways:
As representing function in Cartesian plane
Curve of;As the path of future shadows;The shape of an open orbit (different from a closed elliptical orbit), such as the orbit of a spacecraft during the gravity assisted swing of a planet, or more generally, any spacecraft that exceeds the escape speed of the nearest planet;As a single comet (a travel too fast to return to the solar system) path;As the scattering track of subatomic particles (acting by repulsion rather than attraction, but the principle is the same);In radio navigation, when the distance between two points rather than the distance itself can be determined, etc.
Each branch of the hyperbola has two arms that are straighter (lower curvature) and further extend from the center of the hyperbola.The arm opposite the diagonal, one from each branch, tends to a common line, called the asymptote of the two arms.Therefore, there are two asymptotes whose intersection point is located at the symmetric center of the hyperbola, which can be considered as the mirror point of each branch reflecting to form the other branch.On the curve
In the case of the asymptote, there are two coordinate axes.
Hyperbola shares manyellipseAnalysis attributes such as eccentricity, focus, and pattern.Many other mathematical objects originate from hyperbolas, such asHyperbolic paraboloid(Saddle surface), hyperboloid ("trash can"), hyperbolic geometry (Lobachevsky's famousNon-Euclidean Geometry ）, hyperbolic functions (sinh, cosh, tanh, etc.) and gyroscope vector space (proposed geometry for relativity and quantum mechanics, not Euclid).[2]
Name definition
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We compare the plane with two fixed points Fone，FtwoOf distancedifferenceOfabsolute valueEqual to oneconstant(Constant is 2a, less than | FoneFtwo|) is calledhyperbola;Distance difference between two fixed points in the planeabsolute valueThe path of a point of fixed length is calledhyperbola。
Namely: | | PFone|-|PFtwo||=2a
Definition 1：
planeInside, to twofixed pointOf distanceabsolute valueThe track of a point that is a constant 2a (less than the distance between the two fixed points) is calledhyperbola。The fixed point is called the focus of the hyperbola, and the distance between the two focuses is called the focal length, which is represented by 2c.
Definition 2: distance from a given point and a straight line in the planethanIs the constant e (e>1, that is, hyperbolicEccentricity；The track of a point whose fixed point is not on a fixed line is calledhyperbola。The fixed point is called the focus of hyperbola, and the fixed line is called hyperbolaGuide line。
Definition 3: one plane cut into onecircular cone surface, when the section isconeFacea bus or bus barWhen it is not parallel and does not pass through the vertex of the cone surface, and it intersects both cones of the cone surface, the intersection line is called hyperbola.
In analytic geometry in high school, what I learned is that the center of the hyperbola is at the origin, and the image is symmetric about the x and y axes.Then the hyperbolic equation degenerates to:Ax²+Cy²+F=0
The four definitions above areequivalenceAnd judge the image about the x, y axis according to the front and back positions of the negative signsymmetric。
As can be seen from the image, the hyperbola has two branches.When the focus is on the x-axis, it is left branch and right branch;When the focus is on the y-axis, it is the upper branch and the lower branch.
focus
The two fixed points mentioned in definition 1 are calledfocus, a given point mentioned in definition 2 is also hyperbolicfocus。The hyperbola has two focal points, and the horizontal (vertical) coordinates of the focal point meet c ²=a ²+b ².
Guide line
The given line mentioned in definition 2 is called theGuide line。
The equation of the directrix of the hyperbola is:
(focus on x-axis) or
(focus on y-axis)
Eccentricity
The ratio of the distance from a given point to a given line mentioned in definition 2 is called theEccentricity。
Eccentricity
The hyperbola has two focal points, twoGuide line。(Note: although only one focus and one directrix are mentioned in definition 2, given a focus on the same side, one directrix and eccentricity, two hyperbolas can be obtained simultaneously according to definition 2, and the two hyperbolas are symmetric about the imaginary axis. Therefore, the hyperbolas obtained from the focus on both sides, the directrix and the same eccentricity are the same.)
vertex
The straight line where the hyperbola and its focus line lie has two intersections, which are called the vertices of the hyperbola.
Real axis
The line segment between two vertices is called the real axis of the hyperbola, and half of the length of the real axis is called the semi real axis.
Imaginary axis
Let x=0 in the standard equation to get y ²=- b ². This equation has no real root. To facilitate drawing, draw B on the y-axisone(0, b) and Btwo(0, - b), with BoneBtwoIs an imaginary axis.
Asymptote
There are two hyperbolasAsymptote。Asymptote and hyperbola do not intersect.
The solution to the equation of asymptote is to change the constant on the right side of the standard equation to 0, and then the solution to the asymptote can be obtained by solving the binary quadratic method.
Take the hyperbola with focus on the x-axis as an example, change the equation to
, square the two sides after the item shift
This is the asymptote equation of the hyperbola whose focus is on the x-axis.
Similarly, the asymptote equation of the hyperbola whose focus is on the y-axis is
The right focus of the hyperbola is taken as the pole, and the positive direction of the x-axis is taken as the polar axis to establish a polar coordinate systemPolar equationby
。Where e is the eccentricity of the hyperbola, e>1;
It is called hyperbolic focal distance, that is, the distance from the focus to the corresponding collimation line.
Pay attention to the value of polar angle θ. Because the hyperbola e>1, the denominator will be 0.Solve 1-ecos θ=0, and get cos θ=1/e=a/c. There are two points on [0,2 π) to make the equation hold. In fact, these two angles are exactly the tilt angles of two asymptotes.
If the polar coordinate system is established with the left focus as the pole and the positive direction of the x-axis as the polar axis, then the polar coordinate equation of the hyperbola is
。
Focal radius
The line segment obtained by connecting the focus with any point on the hyperbola is called hyperbolaFocal radius, generally rone、rtwoTo represent the left focal radius and the right focal radius.
The focal radius formula can be derived from the distance formula or the second definition of the conic curve.
Slope of vertex line
Connecting a point (excluding two vertices) on a hyperbola with two verticesSlopeThe product is a fixed value
。Some reference books regard this property as the third definition of hyperbola, that is, a point in the plane
And two fixed points
When the product of the slope of the line of is a fixed value (the fixed value>0), the path of P is hyperbolic (but not including two fixed points).
Since hyperbola often appears in the college entrance examination questions and often combines with asymptote, here are some common geometric properties of hyperbola, especially the properties of asymptote, so that these properties can be quickly used in the questions to solve problems.
Properties of asymptotes
(1) Let the right directrix of the hyperbola intersect an asymptote at P, A is the endpoint of the right branch, and F is the right focus, then OP=OA, OP ∨ PF. The same goes for the left.According to this nature,Make a vertical line of the asymptote through the focus, the vertical foot must be on the guide line, and the three sides of Rt △ OPF are exactly a, b, and c.
Prove that the equation of the right directrix is
, let it and asymptote
To
, so we use the distance formula between two points, OP=a=OA.
At the same time, it is obtained from the definition of slope
, so
。Connect PF for use in △ OPFCosine theoremPF=b, ∠ OPF=90 ° can be obtained.That is, the three sides of Rt △ OPF are exactly a, b, and c.
(2) Draw a parallel line of an asymptote through any point P on the hyperbola, intersect the collimation line with Q, then PQ=PF.
It is proved that, taking the right focus and right guide line as examples, the PM ∨ guide line through P is made at M, and according to the second definition of hyperbola,
So PF=PM * e
According to known conditions, the included angle (or its complement angle) between PQ and PM is exactly the inclination angle of the asymptote, so
, so
。
According to the definition of trigonometric function,
(3) Go through a point P on the hyperbola to make a parallel line of the x (y) axis and intersect the asymptote with A and B, then PA * PB=a ² (b ²).
Proof: Take the parallel line of the x-axis as an example.Set P (xzero,yzero）, the line parallel to the x-axis is y=yzero, intersecting asymptote at
。therefore:
(4) Make the perpendicular PM and PN of two asymptotes through a point P on the hyperbola, then
Proof: according to plane geometry knowledge, ∠ MON and ∠ MPN are complementary, so cos ∠ MPN=- cos ∠ MON
According to the symmetry of hyperbola, the x-axis bisects ∠ MONUniversal formula，
therefore
Set P (xzero,yzero）, using the distance formula from a point to a straight line,
therefore
Note that point P is on the hyperbola, with
, insert the above formula to get the final result.
(5) Let a straight line intersect a hyperbola at two points A and B (the same branch or different branches are allowed), and intersect two asymptotes at two points C and D, then AC=BD.In particular, if the straight line is the tangent of the hyperbola and the tangent point is P, then there is PC=PD.
Proof: Use plane geometry.
① When the straight line is perpendicular to the x-axis, the conclusion is immediately obtained according to the symmetry.
② When the straight line is not perpendicular to the x axis, make a perpendicular to the x axis through A and B. The two perpendicular lines intersect two asymptotes at the four points M, N, R and S.
Get two groups of similar triangles △ ACM →△ BCR and △ ADN →△ BDS
So there is
Multiply the two equations to get
According to property 3, AM * AN=BR * BS=b ², so there is AC * AD=BC * BD
AC*(AB+BD)=BD*(AB+AC)
Reduced AC=BD
When CD is tangent, AB coincides with a point P, and then there is PC=PD, that is, a tangent of the hyperbola intersects two asymptotes at two points, and the distance from the tangent point to these two points is equal.
(6) A tangent of the hyperbola intersects the asymptote at two points A and B, then:
①
Is a fixed value;
② OA * OB is a fixed value.
The above fixed values are independent of the tangent point position.
Proof: Obviously, all straight lines parallel to the x-axis have two intersections with the hyperbola, so they are not tangent lines, that is, the tangent line of the hyperbola must not be parallel to the x-axis.
So let the tangent equation be x=my+n, set up the hyperbolic equation, eliminate x, and get
Tangent and hyperbola have only one intersection point, and the discriminant is 0, so
Simplified n ²=a ² - b ² m ²
From this formula, it can be seen that a ²>b ² m ² (because n must not be 0, if n=0, b ² m ²=a ², making the quadratic term of the above equation 0, the contradiction equation a ² b ²=0 is obtained), so a ± bm ≠ 0.
Simultaneous tangent and two asymptote equations can be solved
If the tangent line and the x-axis intersect at N (n, 0), then
It is a fixed value, independent of the position of the tangent point.
To calculate OA * OB, it is very troublesome to set the distance formula directly, so the vector method can be used.
because
, and
(See property 4 for proof), so
It is a fixed value, independent of the position of the tangent point.
(7) The area of parallelogram OAPB is a fixed value (independent of the position of P) if the parallels of two asymptotes are drawn through any point P on the hyperbola and intersect at two points A and B respectively.
Set P (xzero,yzero）, then the equations of PA and PB can be written as
and
, then we can calculate
Using the area formula of parallelogram,
This property can also be regarded as the deduction of property (5) (6), because it is assumed that the tangent intersection asymptote passing through P is at two points M and N, and from property (5), P is the midpoint of MN.So byThe Theorem of Triangle Median, A and B are the midpoints of OM and ON respectively.Then S △ OAP=1/2 * S △ OMP, S △ OBP=1/2 * S △ ONP.So S △ OAP+S △ OBP=1/2 * (S △ OMP+S △ ONP), that is, S parallelogram OAPB=1/2 * S △ OMN.According to property (6), S parallelogram OAPB=1/2 * ab is a fixed value.
Other properties
Because the conic curve involves almost only the parabola with the focus on the y-axis, the hyperbola will not be examined, but as a supplement, the following properties are still given.
(8) Hyperbola
Any point above
The tangent equation of is
(Note: after the slope is calculated by using the derivation rule of implicit function, the tangent equation is written according to the pointwise oblique form)
(9) Let the intersection of the tangent line and the directrix of the hyperbola at point P be Q, then ∠ PFQ=90 °.The tangents at both ends of the focus chord intersect on the guide line.
(10) If PF1 and PF2 are two focal radii, then the tangent of point P is bisected ∠ F1PF2.Conversely, if the angle between the bisecting radius PF1 and PF2 of a line is known, then the line and the hyperbola are tangent to P.This property is also called hyperbola optical property, that is, the light emitted from a focus, after being reflected by hyperbolaReverse extenderPassing another focus.
(11) If the focus of the ellipse and hyperbola are the same, and the tangents of the ellipse and hyperbola are respectively made at the intersection of the ellipse and hyperbola, then the two tangents are perpendicular.
Mongolian yen issue
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Suppose that two mutually perpendicular tangents of the hyperbola intersect P, then the trajectory of P is a circle (4 intersections with the asymptote are removed).
Set intersection point P (xzero,yzero）, because the tangent of the hyperbola cannot be parallel to the x axis, the other tangent cannot be parallel to the y axis, that is, both tangents have slopes.
Let the tangent equation be y=k (x-xzero)+yzero, simultaneous hyperbola, eliminate y to get
Because the straight line is tangent to the hyperbola, the discriminant is 0, so
Tidied up
The two tangents are perpendicular to each other, and the product of the slope is - 1 according toVeda's theorem, Yes
Tidied up
。
However, it is noted that in the equation of simultaneous tangent and hyperbola at the beginning, the coefficient of the quadratic term cannot be 0, that is
。Substituting this relationship into the quadratic equation of one variable about k, we get
。Therefore, the trajectory of P is a circle that removes the four intersections with the asymptote
This circle is calledMongolian yen, also called outer quasi circle.
be careful:Only when a>b, the equation x ²+y ²=a ² - b ² represents a circle, the eccentricity of the hyperbola
。
Inner quasi circle problem
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The concept of outer quasi circle (Mongolian yen) was introduced above. Now let's study the concept of inner quasi circle.
Let AB be a chord of the hyperbola (A and B can be on the same or different branch), and the angle of the chord to the center O ∠ AOB=90 °, then the distance from O to the straight line AB is a constant regardless of the position of AB.The circle with the constant as the radius and the center O as the center is called the inner quasi circle of the hyperbola.
In order to prove that the distance from O to AB is a constant, we first prove a lemma.
Lemma: if A and B are in hyperbola
On the other hand, OA ∨ OB, then
Is a constant (independent of A and B positions).
Set up with O as the pole, and the non negative half axis of x axis as the polar axisPolar coordinate system, according topolar coordinatesAny point (ρ, θ) on the hyperbola satisfies the transformation relationship with rectangular coordinates
。
As OA ∨ OB, it is advisable to set A (ρone,θ)，B(ρtwo, θ+90 °), substituted into the above equation
The second equation uses trigonometric functionsInduction formulaCos (θ+90 °)=- sin θ and sin (θ+90 °)=cos θ.
According to the definition of polar coordinates,
Is a fixed value.
With lemma, the distance from O to AB can be proved to be a fixed value by using the area method.
Let the distance from O to AB be d, according to the area formula of the triangle
。
Square both sides, get
。
therefore
Is a fixed value.
The equation of the inner quasi circle is
。
Note: opposite to the outer quasi circle,The condition for having an inner quasi circle is
, so the hyperbola can only have one inner and outer quasi circle.In particular,Equiaxed hyperbola(also called right angle hyperbola, satisfying a=b) has neither inner nor outer quasi circle.
This property can be simply remembered as follows: the chord of any tangent line of the quasicircle in the hyperbola cut by the hyperbola, and the angle to the center O is a right angle.
Focal triangle area formula
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Let P be a point on the hyperbola, Fone、FtwoTwo focal points, △ PFoneFtwoIt is called the focus triangle.If ∠ FonePFtwo=θ，
Then S△F1PF2=
。
deduction:
You can set | PFone|=m，|PFtwo|=n，|FoneFtwo|=2c。According to the cosine theorem,
According to the definition of hyperbola, | m-n |=2a, the square of both sides, m ²+n ²=2mn+(2a) ²
Expression substituted into cosine theorem:
So I got
therefore
In addition, if P (xzero,yzero）, then △ PFoneFtwoIt can be seen that the bottom is 2c and the height is | yzero|Triangle of, then
So if you know the coordinates of point P or ∠ FonePFtwoOne of them, you can ask for the other.
hyperbola
If we study this conclusion from a geometric perspective:
As shown in the figure, make the tangent line of hyperbola through P and intersect the right directrix with Z.
Then according to the geometric properties (8) and (9), PZ is an angular bisector, and ∠ PSZ=90 °
But PM ∨ MZ, that is ∠ PMZ=90 °, so the four points of PMZS are in the same circle
, according to the definition of trigonometric function,
, i.e
Example: F1 and F2 are known as hyperbola C: xtwo－ytwo=Left and right focus of 1, point P on C, ∠ FonePFtwo=60 °, the distance from P to x axis is more
Less?
Solution: According to the hyperbolic focus triangle area formula:
S△FonePFtwo=btwo×cot（θ/2）=
If the distance from P to x axis is h, then S △ FonePFtwo=；h=
a key
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Value range
│ x │≥ a (focus on the x-axis) or │ y │≥ a (focus on the y-axis).
Symmetry
It is symmetrical about the coordinate axis and the origin, where it is centrally symmetrical about the origin.
vertex
A（-a，0），A'（a，0）。Meanwhile, AA 'is called hyperbolicReal axisAnd │ AA '│=2a.
B (0, - b), B '(0, b). At the same time, BB' is called hyperbolicImaginary axisAnd │ BB '│=2b.
Fone(- c, 0) or (0, - c), Ftwo(c, 0) or (0, c). FoneIs the left focus of the hyperbola, FtwoRight focus of hyperbola and │ FoneFtwo│=2c
The real axis, imaginary axis and focus are: atwo+btwo=ctwo
Asymptote
Focus on x-axis:
Focus on y-axis:
Eccentricity
First definition: e=c/a and e ∈ (1,+∞)
Second definition: the distance from point P on the hyperbola to the fixed point F │ PF │ and the distance from point P to the fixed line (correspondingGuide line）The ratio of the distance d of is equal to the eccentricity e of the hyperbola.
D point │ PF │/d line (distance from point P to fixed line (corresponding guide line))=e
Focal radius
(Distance from any point P (x, y) on the conic curve to the focus)
Left focal radius: r=│ ex+a │
Right focal radius: r=│ ex-a │
Equiaxed hyperbola
The real axis of a hyperbola is equal to the imaginary axis, that is, 2a=2b and e=√ 2
At this time, the asymptote equation is: y=± x (whether the focus is on the x axis or the y axis)
Conjugate hyperbola
Hyperbolic S'Real axisWhen is the imaginary axis of hyperbola S and the imaginary axis of hyperbola S' is the real axis of hyperbola S, hyperbola S' and hyperbola S are calledConjugate hyperbola。
It is thus proved that,Inverse proportional functionIn fact, it is a form of hyperbola, just another form of hyperbola in the plane rectangular coordinate system.
Hyperbolic inner, upper and outer
The area on both sides of the hyperbola is called hyperbola, then there is xtwo/atwo-ytwo/btwo>1；
On the hyperbola line called hyperbola, there is xtwo/atwo-ytwo/btwo=1；The area between hyperbolas is called hyperbolaxtwo/atwo-ytwo/btwo。
optical properties
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Optical properties of hyperbola
The light emitted from one focus of the hyperbola will converge to the other focus of the hyperbola after being reflected by the hyperbola.The reverse virtual focusing property of hyperbola isAstronomical telescopeWe can also find practical applications in design and other aspects.
