1.
Ellipse: A trajectory whose sum of the distances from a moving point to two fixed points is equal to a fixed length (the fixed length is longer than the distance between the two fixed points) is called an ellipse. Namely: {P|
|PF 1|+|PF2|=2a，
(2a & gt|F 1F2|)} .
2.
Hyperbola: The trajectory of a moving point with a fixed absolute value (the fixed value is less than the distance between two fixed points) is called hyperbola. That is, {p |||| pf1| | pf2 || = 2a,
(2a & lt|F 1F2|)} .
3.
Parabola: A trajectory with the same distance from a moving point to a fixed point and a fixed line is called parabola.
4.
Unified definition of conic section: The locus of a point whose ratio e between the distance to a fixed point and the distance to a fixed line is constant is called conic section. When 0
1 is a hyperbola.
Parametric equation and rectangular coordinate equation of conic curve;
1) straight line
Parameter equation: x=X+tcosθ.
y=Y+tsinθ
(t is the parameter)
Cartesian coordinates: y=ax+b
2) Circle
Parametric equation: x=X+rcosθ
y=Y+rsinθ
(θ is a parameter.
)
Cartesian coordinates: x 2+y 2 = r 2
(r
Is the radius)
3) Ellipse
Parameter equation: x=X+acosθ.
y=Y+bsinθ
(θ is a parameter.
)
Cartesian coordinates (center as origin): x 2/a 2
+
y^2/b^2
=
1
4) hyperbola
Parametric equation: x=X+asecθ.
y=Y+btanθ
(θ is a parameter.
)
Cartesian coordinates (center as origin): x 2/a 2
-
y^2/b^2
=
1
(Opening direction is X axis)
y^2/a^2
-
x^2/b^2
=
1
(Opening direction is Y axis)
5) Parabola
Parametric equation: x = 2pt 2
y=2pt
(t is the parameter)
Cartesian coordinates: y = ax 2+bx+c
(the opening direction is the y axis,
a & gt0
)
x=ay^2+by+c
(the opening direction is the x axis,
a & 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.