The focal radius formula for a conic curve?!

The focal radius formula for a conic curve is as follows:

1) The focal radius formula for the ellipse.

Let m(m,n) be the point of the ellipse x 2 a 2 + y 2 b 2 = 1 (a>b>0) and r1 and r2 are the distances of the point m from the point f (-c,0), f (c,0) respectively, then:

Left focal radius) r = a + em, right focal radius) r = a -em, (e is the eccentricity).

2) The focal radius formula for hyperbola.

The hyperbolic standard equation 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.

Then there is: 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)

3) The focal radius formula for the parabola.

Let the diameter of the parabola be 2p, the parabola equation is y 2=2px(p>0), and c(xo,yo) is a point on the parabola, then the focal radius is |cf|=xo+p/2。

The formula for focal radius in general, and derivation 1The formula for the focal radius of the ellipse is such that m(xo,y0) is the ellipse x2 a2+

A point of y2 b2=1(a>b>0), r1 and r2 are the distances between the point m and the points f1(-c,0), f2(c,0), respectively, then (left focal radius) r1=a+ex0, (right focal radius) r2=a

ex0, where e is the eccentricity. Derivation: r1 mn1 =

r2/∣mn2∣=e

Available: r1=

e∣mn1∣=

e（a^2/

c+x0）=

a+ex0,r2=

e∣mn2∣=

e（a^2/

c-x0）=

a-ex0.In the same way: mf1 =

a+ey0,∣mf2∣=

The focal radius formula for the hyperbola is the focal radius formula when the point p is on the right branch of the hyperbola, (where f1 is the left focal point and f2 is the right focal point) It is derived from the second definition, where a is the length of the real semi-axis, e is the eccentricity, xis the abscissa of the p-point.|pf2|=ex.-

A and only the right branch is remembered, and there is only one minus sign between the left branch and the right branch. If the focus is on the y-axis, only the upper branch is noted.

The radius of the hyperbola over the right focal point r=|a－ex|

The radius of the hyperbola past the left focal point r=|a＋ex|3.The focal radius formula for a parabola is 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 2b2 a, and the focal distance is a2 c-c

The diameter of the parabola is 2p

Parabola y 2 = 2px

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

The line segment formed by the connection from a point to the focal point on a conic curve is called the focal radius of the conic curve. If the point on the conic curve is p, then pf1 and pf2 are their focal radii.

The focal radius of a conic curve is: the distance from any point q on the quadratic curve to the focal point

The concept of focal radius of conic curve is an important concept in conic curve Many problems of solving conic curves are often involved in it, and the use of conic curves to analyze the focal radius of the problem can bring vitality to the solution Therefore, it is very important to master it

Elliptical focal radius: r left = a + x e, r right = a- x e, right hyperbolic focal radius: r left = x e + a, r right = x e- a ( x > 0), left hyperbolic focal radius:

r left = - x e + a), r right = - x e - a) (x < 0), parabolic focal radius: r throw = x + p 2

For example, if F1 is known and F2 is the left and right focus of the ellipse E, the parabola c takes F1 as the vertex and F2 as the focal point, and let p be the intersection point of the ellipse and the parabola, if the eccentricity e of the ellipse E satisfies |pf1| = e | pf2 |, then the value of e is

Solution: Defined by the ellipse |pf1| +pf2 |= 2a, again |pf1| = e | pf2 |，pf2 | 1+ e) = 2a，……

It is also defined by the parabola | pf2 |= x0 + 3c, i.e. x0 = | pf2 | 3c，……

Defined by the ellipse | pf2 |= a- ex0 , which is obtained by | pf2 | = a- e | pf2 |3ec, i.e. | pf2 | 1+ e ) = a + 3ec， …

From 2a = a + 3ec, the solution is obtained.

e=√3/3

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。

Any point on the curve is connected to the focal point of the focal segment of the focal chord, passing the chord path of one focal point. In a chord conic curve (other than a circle) that crosses the focus and is perpendicular to the axis, the chord that crosses the focus and is perpendicular to the axis.

The main focal length formulas of conic curves are:

1. Ellipse focal radius a+ex (left focus), a-ex (right focus), x=a c.

2. Hyperbolic focal radius |a+ex|(Left Focus)|a-ex|(right focus), alignment x=a c.

3. Parabola (y = 2px) focal radius x+p 2 quasi-line x=-p 2.

Chord length = k +1* (x1+x2) -4x1x2 and above is focused on the x-axis, and the y-axis only needs to be replaced by x.

Two. Hyperbola 1, diameter length = 2b a.

2. The focal radius formula (there are 8, it is difficult to make symbols, but it can be directly solved according to the polar coordinate equation, which is faster than the focal radius formula).

3. The area formula of the focal triangle, s pf1f2 = b cot( 2).

Three. The parabola y = 2px (p 0) crosses the straight line of the focal point at a(x1,y1), b(x2,y2) two points, 1, ab = x1 + x2 + p =2p sin is the inclination angle of the straight line ab).

2、 y1*y2 = p_ ，x1*x2 = p_/4。

│fa│ +1/│fb│ =2/p。

4. Conclusion: The circle with ab as the diameter is tangent to the alignment of the parabola.

5. The focal radius formula: fa = x1 + p 2 = p (1-cos).

The length of the line segment connecting a point on a conic curve (including ellipses, hyperbolas, and parabolic lines) with the corresponding focal point is called the focal radius of the conic curve.

Ellipse focal radius.

Let m(x0,y0) be the point of the ellipse x a +y b =1, the focal radii r1 and r2 are the distances between the point m and the points f1(-c,0), f2(c,0), respectively, and e is the eccentricity.

then r1=a+ex0, r2=a-ex0, hyperbolic focal radius.

Let m(x0,y0) be the point of the hyperbola x a -y b = 1, the focal radius r1 and r2 are the distances between the point m and the points f1(-c,0), f2(c,0), respectively, and e be the eccentricity.

The radius of the right focal point r=|ex0-a|

The radius of the left focal point r=|ex0+a|

Parabolic focal radius.

where y = 2px focal radius r = x0 + p 2

The focal radius formulas of conic curves (ellipses, hyperbolas, parabolas) are different on the surface, but their essence is the same, and they are all derived from the second definition, (i.e., the ratio of the distance m from any point of the conic curve m to the focal point f and the distance from m to the corresponding reference is equal to the eccentric rate e).

It's just that the hyperbola has two branches, which have more than the ellipse and do not correspond to the focal radius.

In the standard form of parabola, the constant p directly represents the distance from the focal point to the alignment, and the eccentricity e=1, and when pushing, it is directly represented by p,1.

So the formulas introduced seem to be different on the surface, but the essence is the same. We just need to grasp the essential definition and apply it flexibly.

The focal radius formula when the point p is on the right branch of the hyperbola, (where f1 is the left vertical focus and f2 is the right focus) it is derived from the second definition, where a is the length of the real half-axis, e is the eccentricity, and x. is the abscissa of the p-point.|pf2|=ex。-a

And only the right branch is remembered, and there is only one minus sign between the left branch and the right branch.

If the focus is on the y-axis, only the upper branch is noted.

Infiltration The radius of the right focal point destroyed by hyperbolic fibers r=|a－ex|The radius of the hyperbola past the left focal point r=|a＋ex|

Parabolic focal radius formula.

Parabola r=x+p 2

Diameter: It is a string through which the focal point is perpendicular to the axis, and the focal radius is half diameter.

The diameter of the hyperbola and the ellipse is 2b2 a

The diameter of the parabola is 2p