What is the diameter formula for ellipse and hyperbola?

Alignment: ellipse and hyperbola: x=(a2)c

Parabola: x=p 2

Take y 2=2px as an example).

Focal radius: ellipse and hyperbola: a ex

e is the eccentricity. x is the abscissa of the point, less than 0 takes the plus sign, and greater than 0 takes the minus sign) parabola: p 2 + x

Take y 2=2px as an example).

The above ellipse and hyperbola take the focus on the x-axis as an example.

Chord length formula: Let the slope of the straight line where the string is located be k, then the chord length = root number [(1+k 2)*(x1-x2) 2] = root number [(1+k 2)*(x1+x2) 2-4*x1*x2)].

Using the equation of the straight line and the equation of the conic curve, the unary quadratic equation about x is obtained by eliminating y, x1 and x2 are the two roots of the equation, and x1+x2 and x1*x2 can be obtained by using Veda's theorem, and then substituting the formula to obtain the chord length.

Parabolic diameter = 2p

Parabolic focal chord length = x1 + x2 + p

By connecting the equation of the focal chord with the equation of the conic curve, we obtain a quadratic equation about x by eliminating y, and x1 and x2 are the two roots of the equation.

The formula for the diameter of an ellipse is d=2b a. The ellipse is the sum of the distances from the plane to the fixed points f1 and f2 equal to the constant (greater than |f1f2|The trajectories of the moving point p, f1 and f2 are called the two foci of the ellipse. The mathematical expression is:

pf1|+|pf2|=2a（2a>|f1f2|）。

In mathematics, an ellipse is a curve in a plane around two focal points such that for each point on the curve, the sum of the distances to the two focal points is constant. Therefore, it is a generalization of a circle, which is a special type of ellipse with two focal points at the same position. The shape of an ellipse is represented by its eccentricity, and for an ellipse it can be any number from 0 (the limit case of a circle) to anything that is arbitrarily close but less than 1.

Ellipse Diameter Formula: <>

Ellipse diameter length theorem: the diameter ab of the elliptic shift is the line segment ab obtained by the intersection of the line perpendicular to the major axis and the ellipse by the <> of the focal state.

Derivation process: <>

Solution: <

The path formula is d 2ep (p = distance from focus to alignment).

Focus on the x-axis: |pf1|=a+ex |pf2|=a-ex(f1, f2 are the left and right focus, respectively).

The radius of the ellipse over the right focal point r=a-ex.

The radius of the left focal point r=a+ex.

Focus on the y-axis: |pf1|=a+ey |pf2|=a-ey(f2,f1 are the upper and lower focus, respectively).

The diameter of the ellipse: the distance between the straight line perpendicular to the x-axis (or y-axis) of the focal point and the two intersections of the ellipse a,b, i.e., |ab|=2*b^2/a。

Geometric properties of ellipses

1. Range: The focus is on the x-axis -a x a, -b y b; The focus is on the y-axis -b x b, -a y a.

2. Symmetry: symmetry on the x-axis, symmetry on the y-axis, symmetry on the center of the origin.

3. Vertices: (a,0)(-a,0)(0,b)(0,-b).

4. Eccentricity range: 05. The smaller the eccentricity, the closer it is to the circle, and the larger the ellipse, the flatter the ellipse.

6. Focus (when the center is the origin) :(c,0),(c,0) or (0,c),(0,-c).

Parametric equation for ellipse:Description:

1) The intersection of the major and minor axes of the ellipse is called the center of the ellipse.

3) The eccentricity indicates the degree of flattening of the ellipse, the larger the eccentricity, the flatter the ellipse; When the eccentricity is 0, i.e., a=b, the ellipse is a circle.

The diameter formula is very easy to push. The ellipse is to let x=c, and find the coordinates of y. The elliptic equation is x a + y b = 1, so we get y = b a, and the diameter is the sum of the two lengths of plus and minus, so it is 2b a

HyperbolaThe third definition is that the product of the slope of the change from the moving point in the plane to the two fixed points a1(a,0) and a2(-a,0) is equal to the constant e 2-1The trajectory of the pointIt's called an ellipse or hyperbola. Two of these points are the vertices of an ellipse or hyperbola. When the constant is greater than -1 and less than 0, it is an ellipse; When the constant is greater than 0, it is hyperbola.

The difference in distance from two fixed points (called focal points) is the trajectory of a constant point, and this fixed distance difference is twice that of a.

The third definition of the nature of the curve

The trajectory of the point where the product of the slope of the in-plane moving point to the two fixed points a1(a,0) and a2(-a,0) is equal to the constant e-1 is elliptical or hyperbola. Two of these points are the vertices of elliptic circles or hyperbolas. When 01 is hyperbola.

Conic. The (incomplete) uniform definition of quadratic curves is the trajectory of a point where the quotient of the distance to the fixed point (focal point) and the distance to the fixed line (quasi-line) is the constant e (eccentricity). When e>1, it is a hyperbola, and when e=1, it is a parabola.

When 0 is parallel to the bus of the secondary cone, but not the vertex of the cone, the result is a parabola. When the plane is parallel to the bus of the quadratic cone and passes the vertex of the cone, the result is degenerated into a straight line.

Diameter: A string that passes through the focal point and is perpendicular to the axis on which the focal point is located. This is not an important concept (for conic curves only).

As long as you know and know how to count. In the parabola y 2=2px, let x=p 2 get y=+'p, so the diameter d=2p

In the ellipse, hyperbola x 2A2+'-y 2 b 2=1 in x=c, we get y=+'b 2 a, then d = 2b 2 a. Further analysis shows that d = 2ep (e is the eccentricity and p is the focal parameter--- distance from the focal point to the corresponding alignment. )

Or. The diameter of the ellipse is the length of the line segment that intersects the ellipse with a straight line with a focal point perpendicular to the major axis.

So by substituting x in the elliptic equation to c, we get y1=b 2 a, y2=-b a, so the length of the diameter is y1-y2=2b 2 a

The hyperbolic path formula is also a square of 2b.

The square of the elliptical diameter formula 2b a.

The parabolic path formula is 2p.

The line segment that connects any two points on the ellipse is called the ellipse string, the string that passes through the focus is called the focus chord of the ellipse (so the long axis of the ellipse is also the focus chord), and the focus string perpendicular to the major axis is called the path of the ellipse (the focus chord).