What is the focal radius of an ellipse and what is the formula for the focal radius of an ellipse?

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Let m(xo,y0) be the point of the ellipse x2 a2 + 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), then (left focal radius) r1=a+ex0, (right focal radius) r2=a -ex0, where e is the eccentricity. Derivation: r1 mn1 = r2 mn2 = e yields:

r1= e∣mn1∣= e（a^2/ c+x0）= a+ex0，r2= e∣mn2∣= e（a^2/ c-x0）= a-ex0。Similarly: MF1 = A+EY0, MF2 = A-EY0.

It's a little p on the ellipse

Then the lengths of pf1 and pf2 are focal radius.

where PF1=A+EX and PF2=A-EX

x is the abscissa of point p.

The focal radius of the ellipse: MF1=A+EX0, MF2=A-EX0, and X0 is the abscissa of M.

Derivation of the focal radius formula:Using the second definition of hyperbola, let the hyperbola and its left and right focal points be defined by the second definition: the same is the focal radius formula of the hyperbola with a focus on the x-axis, and the same is the focal radius formula of the hyperbola with a focus on the y-axis.

Among them are the lower and upper foci of the hyperbola. Note: The difference between the hyperbolic focal radius formula and the ellipse focal radius formula is that it has an absolute value symbol, and if you want to remove the absolute value, you need to discuss the position of the point.

Related conclusions. a(x1,y1),b(x2,y2),a,b on the parabola y1=2px, then there is:

When the line ab crosses the focal point, x1x2 = p 4 , y1y2 = p . (When a,b are on the parabola x = 2py, then x1x2 = p and y1y2 = p 4, which can only be true when the line crosses the focus).

Focus chord length: |ab| =x1+x2+p = 2p/[（sinθ）1]=（x1+x2）/2+p。

1/|fa|）+1/|fb|）=2/p；(The length of the long strip is p (1-cos) and the length of the short strip is p (1+cos)).

If OA is perpendicular to ob, AB passes the fixed point m(2p,0).

Ellipse focal radius formula: |pf1|=a+ex0 |pf2|=a-ex0

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

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

The diameter of the ellipse: the distance between the straight line perpendicular to the x-axis (or y-axis) through the focal point and the two intersections of the ellipse a,b, the value = 2b 2 a

The position relationship between the point and the ellipse: the point m(x0,y0) ellipse x 2 a 2+y 2 b 2=1

The dot is inside the circle: x0 2 a 2 + y0 2 b 21

The dot is on a circle: x0 2 a 2+y0 2 b 2=1

The dot is outside the circle: x0 2 a 2+y0 2 b 21

Line vs. ellipse position:

y=kx+m ①

x^2/a^2+y^2/b^2=1 ②

Pushable by x 2 a 2 + (kx+m) 2 b 2 = 1

Tangent = 0 apart 0 no intersection.

Intersect 0 using the chord length formula: a(x1,y1) b(x2,y2).

ab|=d = 1+k^2)|x1-x2| =1+k^2)(x1-x2)^2 = 1+1/k^2)|y1-y2| =1+1/k^2)(y1-y2)^2

Ellipse diameter (definition: chord in a conic curve (other than a circle), passing through the focus and perpendicular to the axis) formula: 2b 2 a

The focal radius of the ellipse is formulated as r1=a+ex , r2=a-ex, where e is eccentricity = c a.

Let m(m,n) be the point of the ellipse (a>b>0), r1 and r2 are the distances between the point m and the points f (-c,0), f (c,0) respectively, then (left focal radius) r = a + em, (right focal radius) r = a -em, where e is the eccentricity. Derivation: R Mn1 = R Mn2 = E.

It can be obtained: R1 = E Mn1 = E(A2 C+M) = A+EM, R2 = E Mn2 = E(A2 C-M) = A-EM.

So: mf1 = a+em, mf2 = a-em.

The focal radius of hyperbola and its application:

1. Definition: The connection 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 hyperbola is known, 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).

The focal radius of a conic curve is the length of the line segment that connects a point on a conic curve (including ellipses, hyperbolas, and parabolas) to the corresponding focal point. It is divided into elliptical focal radius, hyperbolic focal radius, and parabolic focal radius.

The formula for the tilt angle of the focal radius of the elliptical potato forest is =ep (1-cos). 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 that revolves around two focal points such that for each point on the curve, the sum of the distances to the two focal points is constant in the rolling hand. 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 (how it "elongates") is indicated by its eccentricity, and for an ellipse it can be any number from 0 (the limit case of the circle) to anything that is arbitrarily close but less than 1.

The focal radius formula for an ellipse:

Let m(m,n) be the point of the ellipse x 2 a 2 + y 2 b 2 = 1 (a>b>0), r1 and r2 are the distances between the mountain point m and the points f (-c, 0), f (c, 0), then (left focal radius) r = a + em, (right focal radius) r = a -em, where e is the eccentricity.

Derivation: R Mn1 = R Mn2 = E.

It can be obtained: R1 = E Mn1 = E(A2 C+M) = A+EM, R2 = E Mn2 = E(A2 C-M) = A-EM.

So: mf1 = a+em, mf2 = a-em.