A circle is a geometric figure. When a line segment rotates around one endpoint on a plane, the trajectory of the other endpoint is called a circle. By definition, compasses are usually used to draw circles. An ellipse is a point where the sum of the distances from a point to two fixed points on a plane is a certain value. Given two points F 1 and F2 on the coordinate plane, if there is an ellipse with F 1 and F2 as its two focal points, then all points p on the ellipse will satisfy the following conditions: the sum of the distances from p to the two focal points is constant: | PF 1 |+| PF2 | = constant, that is, all points p constitute *. With this definition, we can draw an ellipse easily. First, prepare a line, tie the two ends of the line to one point (these two points are regarded as the two focuses of an ellipse), then pick up a pen, move it from one end of the line to the other, and tighten the line until it reaches the limit. At this time, the two points and the pen will form a triangle, then the wire will start to draw, and the line will be tightened continuously, and finally you can complete an oval figure. It is defined that if the sum of the distances between a moving point and two fixed points on the plane is equal to a fixed length, then the trajectory of this moving point is called ellipse. Assume (note that all assumptions are only for the convenience of deriving elliptic equations) that the moving point is P(x

Y), and the two fixed points are F 1 (? c

0) and F2(c

0), then according to the definition, the trajectory equation of the moving point P satisfies (definition): | PF 1 |+| PF2 | = 2a(a > 0), where 2a is a fixed length. Sort out and simplify the above formula, and get: (a2? c2)x2 + a2y2 = a2(a2？ C2) ① when a > C, set a2 again? C2 = b2, then ① can be further simplified: b2x2+a2y2 = a2b2 ② If an ellipse image is represented in a rectangular coordinate system, then two fixed points in the above definition are defined on the X axis. In the equation, 2a is called the major axis length, 2b is called the minor axis length, and the fixed point is called the focal point, so 2c is called the focal length. In the process of hypothesis, a > C, if you don't assume this way, you will find that you can't get an ellipse. When a = c, the trajectory of this moving point is a circle; When a < C, there is no actual trajectory at all, and its trajectory is called a virtual ellipse. Also note that in the hypothesis, there is another place: a2? C2 = b2. Generally speaking, a circle is a special case of an ellipse. Parabola Parabola is the locus of a point on a plane with the same distance and a fixed straight line that does not pass through the point. This fixed point is called the focus of parabola, and the fixed line is called the directrix of parabola. Parabola is a conic curve. The terms alignment and focusing: see above. Axis: Parabola is an axisymmetric figure, and its axis of symmetry is called axis for short. Vertex: The intersection of a parabola and its axis is called the vertex of the parabola. Chord: The chord of a parabola is a line segment connecting any two points on the parabola. Focus chord: The focus chord of a parabola is the chord passing through the focus of the parabola. Positive focal chord: The positive focal chord of parabola is the focal chord perpendicular to the axis. Diameter: The diameter of a parabola is the locus of the midpoint of a set of parallel chords of the parabola. This diameter is also called the yoke diameter of this set of parallel chords. Principal diameter: The principal diameter of a parabola is the axis of the parabola. Parabola in analytic geometry has four standard equations: y 2 = 2px \ quad \ left (p > 0 \right) (open to the right); Y 2 =-2px \ quadrilateral \ left side (p>0 \ right) (open to the left); X 2 = 2py \ quadrilateral \ left (p>0 \ right) (open); X 2 =-2py \ quadrangle \ left side (p>0 \ right) (opening downward); * y 2 in parabola = 2px \ quad \ left (p&g t; 0 \right), the focus is F \left (\frac{p}{2}

0 \ right), the equation of the directrix L is x =-\ frac {p} {2 }；; * in the parabola y 2 =-2px \ quad \ left (p&g t; 0 \right), the focus is F \left (-\frac{p}{2}

0 \ right), the equation of directrix L is x = \ frac {p} {2 }* in parabola x 2 = 2py \ quad \ left (p&g t; 0 \ right), the focus is F \ left (0

\frac{p}{2} \right), the equation of directrix L is y = \ frac {p} {2 }* y 2 = 2px \ quad \ left in parabola (p&g t; 0 \ right), the focus is F \ left (0

-\frac{p}{2} \right), and the equation of directrix L is y =-\ frac {p} {2 }；; Properties of parabola y2 = 2px 1. Intercept: The intercept of parabola on X axis and Y axis is 0, that is, parabola passes through the coordinate origin, that is, the vertex of parabola. 2. symmetry: parabola is symmetrical about x axis. 3. Scope: Because y = \ pm \ sqrt {2px} \ quad \ left (p > 0 \right), so when x ≥ 0, y has a real value. And because x = \ frac {y 2} {2p}, y can take any real value. When x increases, the absolute value of y also increases, so the parabola extends infinitely up and down on the right side of the y axis. 4. Eccentricity: the ratio of the distance from a point to the focus on a parabola to the distance from the point to the directrix is called eccentricity of the parabola. The eccentricity of parabola is equal to 1. The tangent equation of parabola y2 = 2px is 1. After passing the point p \ left on the parabola y2 = 2px (x _ 1)

The tangent equation of y _ 1 \ right) is y 1y = p(x+x 1). For example, y2 = 4x passes through the point \ left (1

The tangent equation of 2 \ right) is 2y=4 \cdot \frac{x+ 1}{2}, that is, x-y+ 1 = 0. 2. Given that the tangent slope of parabola y2 = 2px is k, its tangent equation is y=kx+\frac{p}{2k}. Hyperbola Hyperbola is the locus of a point on a plane. At this point, the absolute value of the difference between the distances to two fixed points F 1 and F2 is constant. F 1 and F2 are called the focal points of hyperbola, and the distance between the two focal points |F 1F2| is called the focal length. The following is the hyperbolic standard equation in the plane rectangular coordinate system: \ frac {x2} {a2}-\ frac {y2} {B2} =1.

Reference: Wikipedia

Circle: two points A and B are fixed on the circumference, and a certain ratio R is fixed. All points c satisfying the condition AC/BC=R form a circular ellipse. Definition: The locus of a point whose sum of the distances between two fixed points F 1 and F2 on a plane is equal to a constant (greater than ||| f1F2 |||) is called an ellipse, these two fixed points are called the focus of the ellipse, and the distance between the two focuses is called the focal length.

Is the * * * value of the point on the same plane whose distance from the fixed point (focus) is equal to the distance from the alignment (directrix); Quadratic equation represents parabola (including quadratic equation of X and quadratic equation of Y); Hyperbola: The absolute value of the distance difference from a fixed point to 2 is equal to the trajectory of a point with a fixed length less than the fixed point. If the ratio of the distance from a moving point to a fixed point on a plane to the distance to a straight line is a constant greater than 1 (the fixed point is not on a fixed straight line),

Then the trajectory of the moving point is a hyperbola.

Reference: Baidu