Hyperbola refers to the locus of a point whose absolute value of the distance difference between two points on a plane is constant, and can also be defined as the locus of a point whose distance ratio to a fixed straight line is constant greater than 1. Hyperbola is a kind of conic curve, that is, the intersection of a conical surface and a plane. Hyperbola can also be regarded as an inverse proportional function under some affine transformation. This catalogue defines the important concepts and properties of hyperbolic standard equations. Simple geometric properties of hyperbola of bifurcation focus directrix eccentricity vertex asymptote 2. Symmetry: 3, vertex: 4, asymptote: 5, eccentricity: 6, hyperbola focal radius formula 7, equilateral hyperbola 8, yoke hyperbola 9, directrix: 10, path length: 12, chord length formula: 13. The area formula of the inner, upper and outer triangles of hyperbola The hyperbolic parameter equation defines the standard equation of hyperbola. Important concepts and properties The simple geometric properties of hyperbola of branch focus directrix eccentricity vertex asymptote II. Symmetry: 3. Vertex: 4. Asymptote: 5. Eccentricity: 6. Hyperbolic focal radius formula 7. Equilateral hyperbola 8. * * Yoke hyperbola 9. Directrix: 10, path length: 12, chord length formula: 13. The hyperbolic parameter equation of the inner, upper and outer triangle area formulas of hyperbola is expanded and edited. Definition of this paragraph: We call the trajectory whose absolute value is equal to a constant from two fixed points F 1 and F2 on the plane a hyperbola. Definition: 1: On the plane, the locus of a point whose absolute value is constant (less than the distance between two fixed points [1]) is called hyperbola. The fixed point is called the focus definition of hyperbola. 2. The locus of a point on a plane whose distance ratio from a given point to a straight line is greater than 1 is constant is called hyperbola. The fixed point is called the focus of hyperbola, and the fixed line is called the directrix of hyperbola. Definition 3: Plane cutting conical surface. When the section is not parallel to the generatrix of the conical surface and intersects with the two conical surfaces of the conical surface, the intersection line is called hyperbola. Definition 4: In the plane rectangular coordinate system, when the binary quadratic equation f (x, y) = ax 2+bxy+cy 2+dx+ey+f = 0 meets the following conditions, it is hyperbolic. 1.a, b and c are not all zero. 2.B 2-4ac > 0.3.A 2+B 2 = C 2 in high school analytic geometry, we learn that the center of the hyperbola is at the origin, as if it were symmetrical about x and y, and then the hyperbolic equation degenerates into: x 2/a 2-y 2/b 2 = 1. The above four definitions are equivalent, and it is judged that the image is symmetrical about X and Y according to the position before and after construction. Edit the standard equation of this hyperbola 1. When the focus is on the X axis, it is: x2/a2-y2/b2 =12. When the focus is on the Y axis, it is: y 2/a 2-x 2/b 2 =1. Edit this paragraph. A hyperbola has two branches. The two given points mentioned in the focus definition 1 are called the focus of hyperbola, and a given point mentioned in definition 2 is also the focus of hyperbola. A hyperbola has two focuses. The given straight line pointed by the directrix in definition 2 is called the eccentricity of hyperbola. The ratio of the distance from a given point to a given straight line mentioned in definition 2 is called the eccentricity of hyperbola. A hyperbola has two focuses and two directrix. (Note: Although only one focus and one directrix are mentioned in Definition 2. But given a focus, a directrix and eccentricity on the same side, according to definition 2, two branches of hyperbola can be obtained at the same time, and the focus, directrix and hyperbola obtained by eccentricity on both sides are the same. The intersection of the vertex hyperbola and the line connecting the two focal points is called the vertex of the hyperbola. Asymptote hyperbola has two asymptotes. Edit the simple geometric property of this hyperbola 1, and the value range of a point on the trajectory: │x│≥a (focusing on the X axis) or │y│≥a (focusing on the Y axis). 2. Symmetry: Symmetry about coordinate axis and origin. 3. Vertex: A(-a, 0), A'(a, 0). At the same time AA' is called the real axis of hyperbola and │AA'│=2a. B(0，-b)，B'(0，B)。 At the same time, BB' is called the imaginary axis of hyperbola and bb' = 2b.f1(-c, 0) F2 (c, 0). F 1 is the left focus of hyperbola, F2 is the right focus of hyperbola, and │F 1F2│=2c has: A is the real axis, imaginary axis and focus. 1 stands for hyperbola. Where p is that distance from the focus to the directrix and θ is the angle between the chord and the X axis. Let 1-ecosθ=0 to find θ, where θ is the inclination of the asymptote. θ=arccos( 1/e) makes θ=0, which leads to ρ = EP/ 1-E, and X = ρ COS θ = EP/ 1-E makes θ=PI, which leads to ρ = EP/1+e. Find the abscissa of their midpoint (the abscissa of the hyperbola center) x = [(EP/1-e)+(-EP/1+e)]/2 (please simplify) The straight line ρ cos θ = [(EP/1-e)+(-EP/). Rotate the line clockwise by π/2-arccos( 1/e) to get the asymptote equation. Let the rotated angle be θ′, then θ′ = θ-[pi/2-arccos (1/e)], then θ = θ′+[pi/2-arccos (1/e)] is substituted into the above formula: ρ cos {θ′+[pi/e]. 2 namely: ρ sin [arccos (1/e)-θ′] = [(EP/1-e)+(-EP/1+e)]/2 Now you can use θ instead of θ′ in the formula to get the equation: ρ sin [arccos (+e)] It is proved that if the point on the hyperbola x 2/a 2-y 2/b 2 =1is the point on the first quadrant of the asymptote, then y = (b/a) √ (x 2-a 2) (x > a) is because x 2-a 2.