I. Equations and properties of conic curves:
1) ellipse
Literal language definition: the ratio of the distance between a moving point, a fixed point and a straight line in a plane is a normal number e less than 1. The fixed point is the focus of the ellipse, the fixed line is the directrix of the ellipse, and the constant e is the eccentricity of the ellipse. ?
Standard equation:?
1. Elliptic standard equation with the center at the origin and the focus at the X axis: (x 2/a 2)+(y 2/b 2) = 1.
Where a>b>0, C>0 and C 2 = A 2-B 2.
2. The standard equation of an ellipse with the center at the origin and the focus at the Y axis: (x 2/b 2)+(y 2/a 2) =1.
Where a>b>0, C>0 and C 2 = A 2-B 2.
Parametric equation:
X=acosθ？ Y=bsinθ？ (θ is a parameter? Let the abscissa be acosθ, and the ellipse can be a circle after telescopic transformation due to the consideration of conic curve. At this time, c=0, and acos θ of the circle = r)
2) hyperbola
Literal language definition: the ratio of the distance between a moving point, a fixed point and a straight line in a plane is a constant e greater than 1. The fixed point is the focus of hyperbola, the fixed line is the directrix of hyperbola, and the constant e is the eccentricity of hyperbola. ?
Standard equation:?
1. Hyperbolic standard equation with the center at the origin and the focus on the X axis: (x 2/a 2)-(y 2/b 2) = 1?
Where a>0, B>0 and C 2 = A 2+B 2.
2. Hyperbolic standard equation with the center at the origin and the focus on the Y axis: (Y 2/A 2)-(X 2/B 2) = 1.
Where a>0, B>0 and C 2 = A 2+B 2.
Parametric equation:
x=asecθ？ y=btanθ？ (θ is a parameter? )?
3) Parabola
Standard equation:?
1. Parabolic standard equation with vertex at origin, focus on X axis and opening to the right: y 2 = 2pxwhere? p & gt0
2. Parabolic standard equation with vertex at origin, focus on X axis and opening to the left: y 2 =-2pxwhere? p & gt0
3. Parabolic standard equation with vertex at origin and focus on Y axis: X 2 = 2py, where? p & gt0
4. Parabolic standard equation with vertex at origin, focus on Y axis and downward opening: x 2 =-2py, where? p & gt0
parameter equation
x=2pt^2？ y=2pt？ (t is a parameter)? T= 1/tanθ(tanθ is the slope of the straight line determined by the point on the curve and the coordinate origin). In particular, t can be equal to 0?
Cartesian coordinates
y=ax^2+bx+c？ (the opening direction is the y axis,? a & lt& gt0? )? x=ay^2+by+c？ (the opening direction is the x axis,? a & lt& gt0? )?
The unified polar coordinate equation of conic curve (quadratic noncircular curve) is
ρ=ep/( 1-e×cosθ)？ Where e stands for eccentricity and p is the distance from the focus to the directrix.
Second, the focal radius.
The distance from any point on a conic curve to the focal point is called the focal radius. ?
The left and right focal points of a conic curve are F 1 and F2, and any point on it is P(x, y), then the focal radius is:?
Ellipse |PF 1|=a+ex? |PF2|=a-ex？
Hyperbolic? P in the left branch, | pf 1 | =-a-ex? |PF2|=a-ex？
P is in the right branch, |PF 1|=a+ex? |PF2|=-a+ex？
P in the next branch, |PF 1|=? -Huh? |PF2|=a-ey？
P in the upper branch, |PF 1|=? a+ey？ |PF2|=-a+ey？
Parabolic? |PF|=x+p/2？
Third, the tangent equation of conical curve?
Tangent equation of point P(x0, y0) on conic curve.
X0x instead of x 2, y0y instead of Y2; Replace x with (x0+x)/2 and y with (y0+y)/2.
Namely ellipse: x0x/a2+y0y/B2 =1;
Hyperbola: x0x/a2-y0y/B2 =1;
Parabola: y0y=p(x0+x)
Fourth, the focal length.
The distance p from the focus of the conic to the directrix is called the focal length or the focal parameter of the conic. ?
Focal length of ellipse: p = (b 2)/c?
Focal length of hyperbola: p = (b 2)/c?
Quasi-focal length of parabola: p
Verb (abbreviation for verb) diameter
In a conic curve, the chord passing through the focal point and perpendicular to the axis becomes the diameter.
Ellipse diameter: (2b 2)/a?
The path of hyperbola: (2b 2)/a?
Path of parabola: 2p
Comparison of conic properties of intransitive verbs
See the figure below:
Seven, the midpoint chord of the conic curve.
Given that a point on a conic curve is the midpoint of a chord of the conic curve, how to find the equation of this chord?
1. simultaneous equation method.
The equation of chord is established in the way of point inclination (if there is no slope, it needs to be considered separately), and the quadratic equation of X and the quadratic equation of Y are obtained simultaneously with the conic equation. The expression of the sum of the two is obtained by Vieta theorem, and the equation of the chord is obtained by the specific value of the sum of the two in the midpoint coordinate formula. ?
2. Point difference method, or substitution point subtraction.
Set the coordinates (x 1, y 1) and (x2, y2) at both ends of the chord, substitute them into the equation of conic curve, subtract the two equations, and get [(x 1+x2)]/(a) by using the square difference formula. The value of the slope can be obtained from the slope of (y 1-y2)/(x 1-x2). (Pay attention to the problem of discriminant when using)