I. Elliptic equation.
1. The first definition of the elliptic equation:
| PF₁|+| PF₂|=2a>| The F₁F₂ | equation is elliptic
| PF₁|+| PF₂|=2a<| F₁F₂|无轨迹
| PF₁|+| PF₂|=2a=| F₁F₂ | line segments with F₁, F₂ as the endpoint
(1) (1) Standard equations for ellipses:
i. The center is at the origin and the focus is on the x-axis: x²/a²+y²/b²=1 (a>b>0). ii. The center is at the origin and the focus is on the y-axis: y²/a²+x²/b²=1(a>b>0).
(2) General equation: Ax² + By² = 1 (A>0, B>0)
(3) The standard parameter equation for an ellipse: x²/a²+y²/b²=1 The parametric equation for x=acosθ.y=bsinθ. (one quadrant θ should belong to 0<θ<π/2).
⑵①顶点：(±a,0)(0,±b)或(0,a±)(±b,0)
(2) Axis: axis of symmetry: x-axis, y-axis; major axis length 2a, short axis length 2b
(3) Focus: (-c,0)(c,0) or (0,-c)(0,c)
④焦距：| F₁F₂|=2c,c=√(a²-b²)
(5)准线:x=±a²/c或y=±a²/c
(6)离率:e=c/a(0<e<1)
(7) Focus radius:
i. Let P(x0.y0) be a point on the ellipse x²/b²+y²/a²=1, F₁, F₂ is the left and right focus, then the | PF₁|=a+ex0,| PF₂|=a-ex0=> can be deduced by the second definition of the elliptic equation.
ii. Let P(x0,y0) be a point on the ellipse x²/b²+y²/a²=1(a>b>0), F₁, F₂ is the upper and lower focus, then the | PF₁|=a+ey0,| PF₂|=a-ey0=> can be deduced by the second definition of the elliptic equation.
From the second definition of the ellipse, it can be seen that |pF₁| = e(x0+a²/c) = a + ex0 (x0<0), |pF₂| = e(a²/c-x0) = ex0-a(x0>0) boils down to "left plus right minus".
Note: Derivation of the elliptic parametric equation: the trajectory of the equation → the equation to obtain N(acosθ,bsinθ) is elliptic.
(8) Diameter: The string perpendicular to the x-axis and passing through the focus is called the passage. Coordinates: d=2b²/a²(-c,b²/a) and (c, b²/a)
(3) Equations of the elliptic system of co-eccentrifuge: the eccentricity of the ellipse x²/a²+y²/b²=1 (a>b>0) is e=c/a (c=√ (a²-b²)), the equation x²/a²+y²/b²=t (t is the parameter greater than 0, the eccentricity of a>b>0) is also e=c/a We call this equation the elliptic equation of the co-eccentricity.
(5) If P is a point on an ellipse: x²/a²+y²/b²=1. F₁, F₂ is the focal point, if ∠F₁PF₂=θ, then △F₁PF₂ has an area of b²tanθ/2 (with the cosine theorem and | PF₁|+| PF₂|=2a available). If it is hyperbolic, the area is b²·cotθ/2.
Second, hyperbolic equations.
1. The first definition of a hyperbola:
|| PF₁|-| PF₂|| =2a<| The F₁F₂ | equation is hyperbolic
|| PF₁|-| PF₂|| =2a>| F₁F₂|无轨迹
|| PF₁|-| PF₂|| =2a=| F₁F₂ | a ray with F₁, an endpoint of F₂
(1) (1) Hyperbolic standard equations: x²/a²-y²/b²=1(a,b>0), y²/a²-x²/b²=1(a,b>0). General equation: Ax² + Cy²=1 (AC<0).
⑵①i. The focus is on the x-axis:
Vertices: (a,0), (-a,0) ; Focus: (c,0), (-c,0); Quasilinear equation x=±a²/c; Asymptote equation: x/a±y/b=0 or x²/a²-y²/b²=0.
ii. Focus on the y-axis:
Vertices: (0,-a), (0,a). Focus: (0,c), (0,-c). Quasiline equation: y=±a²/c. Asymptote equation: y/a±x/b=0 or y²/a²-x²/b²=0, Parametric equations: {x=secθ,y=btanθ or {x=btanθ,y=asecθ .
(2) Axis x, y is the symmetrical axis, the real axis length is 2a, the imaginary axis length is 2b, and the focal length is 2c.
(3) Eccentricity e= c/a.
(4) The crosshairs are 2a²/c (the distance between the two collimators); the diameter is 2b²/a.
(5)参数关系c²=a²+b²,e=c/a.
(6) Focus radius formula: for hyperbolic equation x²/a²-y²/b²=1 (F₁, F₂ are the left and right focal points of the hyperbola or the upper and lower focal points of the hyperbola, respectively)
The principle of "long plus short"
(3) Isometric hyperbolic: hyperbolic x²-y²=±a² is called isometric hyperbolic, and its asymptotic equation is y=±x, eccentricity e=√2.
(4) Conjugate hyperbolic: the imaginary axis of the known hyperbolic curve is the real axis, and the real axis is the hyperbolic curve of the imaginary axis, which is called the conjugate hyperbolic curve of the known hyperbolic curve.x²/a²-y²/b²=λ and x²/a²-y²/b²=-λ are conjugated hyperbolic curves, and they have a common asymptotic line: x²/a²-y²/b²=0.
(5) Hyperbolic equations for co-asymptotic lines: x²/a²-y²/b²=λ(λ≠0) The asymptotic equation is x²/a²-y²/b²=0 If the asymptote of the hyperbolic is x/a±y/b=0, its hyperbolic equation can be set to x²/a²-y²/b²=λ(λ≠0).
For example, if a hyperbolic asymptote is y=1/2x and the solution of p(3,-1/2) is passed: the equation for the hyperbolic is:
Solution: The equation for the hyperbolic curve is: x²/4-y²=λ(λ≠0), substituting (3,-1/2) gives x²/8-y²/2=1.
(6) The position relationship between straight lines and hyperbolic curves:
Area (1): no tangents, 2 straight lines parallel to the asymptote line, total 2;
Area (2): that is, the fixed point is on the hyperbolic curve, 1 tangent line, 2 lines parallel to the asymptote line, a total of 3;
Area (3): 2 tangent lines, 2 lines parallel to the asymptote line, a total of 4;
Area (4): that is, the fixed point is on the asymptote line and is not the origin, 1 tangent line, 1 line parallel to the asymptote line, a total of 2;
Area (5): i.e. over the origin, no tangent, no straight line parallel to the asymptote.
Summary: Over-fixed point for straight lines and hyperbolic curves have and only one intersection point, the number of straight lines that can be made may be 0, 2, 3, 4.
(2) If the line and the hyperbolic line have an intersection point, and the intersection point is two, the slope of the line can be determined by substitution △ method and asymptote intersection and the sum of the two roots and the product of the two roots are the same.
(7) If P is in the hyperbolic x²/a²-y²/b²=1, then the common conclusion 1: the distance from P to the focal point is m = n, and the distance ratio of P to the two quasi-lines is m..n.
简证:d₁/d₂=| PF₁|/e/| PF₂|/e = m/n.
Common conclusion 2: The distance from one focal point of the hyperbolic curve to another asymptote is equal to b.
Third, the parabolic equation.
3. Set p>0, the standard equations, types, and geometric properties of parabolas:
注：①ay²+by+c=x顶点((4ac-b²)/4a-b/2a).
(2) y² = 2px(p≠0) then the focus radius | PF|=|x+P/2|; x²=2py(p≠0) has a focal radius of | PF|=|y+p/2|.
(3) The diameter is 2p, which is the shortest of all strings that pass the focus.
(4) The parametric equation for y²=2px (or x²=2py) is {x=2pt², y=2pt (or {x=2pt, y=2pt²) (t is the parameter).
Fourth, the unified definition of the conic curve:
4. Uniform definition of conic curve: the trajectory of a point in the plane to the fixed point F and the fixed line ι with a ratio of constant e.
When 0<e<1, the trajectory is elliptical;
When e=1, the trajectory is parabolic;
When e>1, the trajectory is hyperbolic;
When e=0, the trajectory is round (e=c/a, when c=0, a=b).
5. The conic curve equation is symmetrical. For example, the standard equation for an ellipse is symmetrical about the origin of a straight line at the origin and the intersection of a hyperbolic curve.
Because of the symmetry, if you want to prove AB=CD, that is, the midpoint of AD and BC coincides.
Note: Standard equations and geometric properties of ellipse, hyperbolic, parabolic
1. Other forms and corresponding properties of the standard equations for ellipse, hyperbolic and parabolic.
2. Isometric hyperbolic
3. Conjugate hyperbolic
5. Focal coordinates and quasilinear equations of the equations y²=ax and x²=ay.
6. Hyperbolic equations of the common asymptote.
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