Ellipse, hyperbola, parabola, alignment, through, focal radius, chord length, over focal chord lengt

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

Parabola: x=p2 (take y2=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 = 2 px as an example) above the ellipse and hyperbola with 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)] The equation of the straight line is connected with the equation of the conic curve, and the unary quadratic equation about x is obtained by eliminating y, and 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 the chord length can be obtained by substituting the formula.

Parabolic diameter = 2p

The length of the parabolic focal chord = x1 + x2 + p The equation of the focal chord is connected to the equation of the conic curve, and the unary quadratic equation about x is obtained by subtracting y, and x1 and x2 are the two roots of the equation.

The distance from any point on the ellipse to the focal point is called the focal radius.

The focal radius equation for the 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).

Alignments: Ellipses and hyperbolas: x=(a2) c (focus on x-axis), y=(a2) c (focus on y-axis).

Parabola: x=p2 (with x-axis as the focal point).

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)].

The formula for the diameter of the ellipse is: 2b 2 a

Parabolic diameter = 2p

The length of the parabolic focal chord = x1 + x2 + p The equation of the focal chord is connected to the equation of the conic curve, and the unary quadratic equation about x is obtained by subtracting y, and x1 and x2 are the two roots of the equation.

(1) When encountering midpoint string problems, the "Vedic theorem" or the "spread method" are commonly used

Vedic theorem: "I won't go into details, but I will focus on the spread method.

2) Use the spread method for mid-point string problems.

The midpoint chord problem generally uses the spread method to find the slope of the straight line.

Taking an ellipse as an example, the elliptic equation x 2 a 2 + y 2 b 2 = 1, (a>b>0).

Let the line l and the ellipse intersect a(x1,y1),b(x2,y2) and the midpoint n(x0,y0).

x1^2/a^2+y1^2/b^2=1

X2 2 Manuscript solution A 2 + Y2 B 2 = 1

Subtract from the two equations (x1+x2)(x2-x1) a 2+(y2+y1)(y2-y1) b 2=0

x1+x1=2x0,y1+y2=2y0

kab=(y2-y1)/(x2-x1)=-b^2* x0/(a^2* y0)

ab equation y-y0=-b 2* x0 (a 2* y0)(x-x0).

Using the analogy of Li Jingzao's method, we can find the slope of the point chord in the hyperbola, which is b 2* x0 (a 2* y0).

The midpoint chord slope of the parabola p y0

The parabolic focal chord length formula is 2p sina 2.

Let the parabola be y 2=2px(p>0), the equation for the string line through the focal point f(p 2,0) is y=k(x-p 2), and the straight line intersects the parabola at a(x1,y1),b(x2,y2).

The simultaneous equation yields k 2(x-p 2) 2=2px, and k2x 2-p(k 2+2)x+k 2p 2 4=0. So, x1+x2=p(k 2+2) k 2.

Defined by the parabola, the distance from af=a to the alignment x=-p 2 x1+p 2, bf = x2+p 2. So:

ab=x1+x2+p=p(1+2/k^2+1)=2p(1+1/k^2)=2p(1+cos^2/sin^2a)=2p/sin^2a。

The nature of the parabolic focal string.

Two tangents at the ends of the focal chord intersect on the alignment, and the intersection is perpendicular to the focal chord. Conversely, if you cross any point on the alignment to make two tangents of a conic curve, the line connecting the two tangents will pass through the focal point.

The relationship between a circle with a focal chord diameter and the corresponding quasi-starvation: ellipse - separation; hyperbola – intersecting; Parabola – tangent.

Ellipse: 1) Focus strings: a(x1,y1), b(x2,y2), ab is the focus chord of the ellipse, m(x,y) is the midpoint of ab, then l=2a 2ex

2) Let the straight line: intersect with the ellipse at p1(x1,y1),p2(x2,y2), and the slope of p1p2 is k, then |p1p2|=|x1-x2|(1+k) or |p1p2|=|y1-y2|√(1+1/k²）

Hyperbola: 1) Focus strings: a(x1,y1), b(x2,y2), ab is the focus chord of the hyperbola, m(x,y) is the midpoint of ab, then l=-2a 2ex

2) Let the straight line: intersect with the hyperbola at p1(x1,y1),p2(x2,y2), and the slope of p1p2 is k, then |p1p2|=|x1-x2|(1+k) or |p1p2|=|y1-y2|√(1+1/k²）

Parabola: 1) Focus string: It is known that the parabola y = 2px, a(x1, y1), b(x2, y2), ab is the focus string of the parabola, then |ab|=x1+x2+p or |ab|=2p/(sin²h）

2) Let the straight line: intersect with the parabola at p1(x1,y1),p2(x2,y2), and the slope of p1p2 is k, then |p1p2|=|x1-x2|(1+k) or |p1p2|=|y1-y2|√(1+1/k²）

The focal chord is made up of two focal radii on the same straight line. The focal chord length is the sum of these two focal radius lengths. The straight line of the elliptic focus f intersects the ellipses at two points A and B, and denotes q=a 2 c-c, which is the focal distance and e is the eccentricity.

Order|fe|=m，|ed|=n, then m+n=|fd|。If and only if, take |cd|The minimum value is 2a. Theorem 1 (principle of polarity theory), if the pole of point p passes through point q, then the pole line of point q also passes through point p.

Ellipse: 1The radius of the right focal point r=a-ex

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

3.The radius of the over-focus r=a-ey

4.The radius of the lower focal point r=a+ey

Extended Resources:

Hyperbola. The focal radius of hyperbola and its application:

1: Definition: The line segment between any point p and the hyperbolic focus on the hyperbola is called the focal radius of the hyperbola.

2 The standard equation of the hyperbola is known x 2 a 2-y 2 b 2 = 1, and f1 is the left focus, f2 is the right focus, and e is the eccentricity of the hyperbola.

Always say: pf1 =|ex+a)| pf2│=|ex-a)|(for any x).

Specifically: the point p(x,y) is on the right branch.

pf1│=ex+a ；│pf2│=ex-a

Point p(x,y) on the left branch.

pf1│=-ex+a) ；pf2│=-ex-a)

Parabola. Parabola r=x+p 2

Diameter: A chord in a conic curve (divided by a circle) that passes through the focal point and is perpendicular to the axis.

The diameter of the hyperbola and the ellipse is 2b 2 a and the focal distance is a c-b c = c

a²－b²=c²

The diameter of the parabola is 2p

The parabola y 2=2px (p>0), c(xo,yo) is a point on the parabola, and the focal radius is |cf|=xo+p/2.