In a plane, the trajectory to a point with the same distance from a fixed line is called parabola. The fixed point is called the focus of parabola, and the fixed line is called the directrix of parabola.

A parabola refers to the locus of a point on a plane with the same distance from a fixed point F (focus) and a fixed line L (directrix). It has many representations, such as parameter representation and standard equation representation. It plays an important role in optics and mechanics. Parabola is also a kind of conic curve, that is, the curve obtained by the intersection of a conic surface and a plane parallel to the generatrix. Parabola can also be regarded as quadratic function image under proper coordinate transformation.

Mathematically, a parabola is a plane curve with mirror symmetry and a roughly U-shaped orientation (if it is different, it is still a parabola). It can be applied to any number of seemingly different mathematical descriptions and can be proved to be exactly the same curve.

The description of parabola includes a point (focus) and a line (directrix). The focus is not on alignment. A parabola is the locus of a point on a plane equidistant from the directrix and the focus. Another description of parabola is that as a conical section, it is formed by the intersection of a conical surface and a plane parallel to the generatrix of the cone. The third description is algebra.

Parabola is an axisymmetric figure, and the straight line perpendicular to the directrix and passing through the focus (that is, the straight line decomposed in the middle of parabola) is the "symmetry axis" of parabola. The point on the parabola that intersects the axis of symmetry is called the "vertex" and is the sharpest point of the parabola. The distance between the vertex and the focus measured along the axis of symmetry is the "focal length". A "straight line" is a parallel line of a parabola that passes through the focus. A parabola can be opened up, down, left, right or any other direction. Any parabola can be repositioned, repositioned to accommodate any other parabola-that is, all parabolas are geometrically similar.

Parabolas have the property that if they are made of materials that reflect light, the light that propagates parallel to the symmetry axis of the parabola and hits its concave surface is reflected to its focus, regardless of where the parabola is reflected. On the contrary, the light generated from the point light source at the focus is reflected as a parallel ("collimated") beam, so that the parabola is parallel to the axis of symmetry. Sound and other forms of energy will have the same effect. This reflection property is the basis of many practical applications of parabola.

Parabolic has many important applications, from parabolic antenna or parabolic microphone to reflector of automobile headlights to design ballistic missiles. They are often used in physics, engineering and many other fields.

* * * Similarities:

① The origin is on a parabola, the eccentricity e is 1 ② the axis of symmetry is the coordinate axis;

③ The directrix is perpendicular to the axis of symmetry, the vertical foot and the focus are symmetrical to the origin respectively, and their distance from the origin is equal to 1/4 of the absolute value of the linear coefficient.

Difference:

① When the axis of symmetry is the X axis, the right end of the equation is 2px and the left end of the equation is y^2；; ; When the symmetry axis is the Y axis, the right end of the equation is 2py and the left end of the equation is x^2；; ;

(2) When the opening direction is the same as the positive semi-axis of X axis (or Y axis), the focus is on the positive semi-axis of X axis (Y axis), and the right end of the equation takes a positive sign; When the opening direction is the same as the negative semi-axis of X (or Y), the focus is on the negative semi-axis of X (or Y), and the right end of the equation takes a negative sign.

Tangent equation

The tangent equation at the point (x0, y0) on the parabola y2=2px is:

. The equation of parabola y2=2px with focal slope k is y=k(x-p/2).

Eccentricity: e= 1 (constant is a constant value, which is the ratio of the distance from a point on a parabola to the directrix and the distance from the point to the focus).

Focus: (P/2,0)

Alignment equation l:x=-p/2

Vertex: (0,0)

Diameter: 2p; Definition: A chord in a conic curve (except a circle) that passes through the focal point and is perpendicular to the axis.

Domain: for parabola y 1=2px, p>0, the domain is x≥0, p.

Range: for parabola y 1=2px, range is r; for parabola x 1=2py, p>0, the range is y≥0, p.

Straight line, focus: A parabola is the locus of a point on a plane, and its distance to a point is equal to the distance to a 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.

Axis: Parabola is an axisymmetric figure, and its axis of symmetry is called axis for short.

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.

A parabola is a curve in which an object is thrown and falls on the distant ground.

I hope it can help you solve the problem.