Where is the analytic geometry ellipse master? 15

1) When m=-1, a is (0,1) and b is (,. Simultaneous y=kx and x 2+4·y 2=4, x1+x2=0, x1*x2=(-4) (1+4k 2), y1+y2=0, y1*y2=(-4k 2) (1+4k 2), and then use the chord length formula to get cd as 4*(1+k2) (the area of the triangle acd and the triangle bcd can be calculated by using the distance from a and b to y=kx, and the sum of the areas is the quadrilateral area, and the final expression is.

s=(, you can start by using the derivative or squaring to find the s maximum.

2) First synthesize the straight line l:x=my+1 and the elliptic equation, use Vedica's theorem to obtain the sum and product of the ordinates and abscissa of a and b, at the set point m(x,0), a is (xa,ya), and b is (xb,yb), you can express the slope of the straight line, and use the condition that the product of the slope of the straight line ma and the straight line mb is fixed, because the denominator is yayb, and the numerator is xaxb-x(xa+xb)+x 2, so you can find a way to make the molecular molecule of this formula exactly be a constant multiple of the real number of the denominator, and finally obtain. x=0

Summary. Ellipsical geometry is Riemannian geometry. Geometry on Riemann manifolds.

The theory of geometry proposed by the German mathematician Riemann in the mid-19th century. Riemann's inaugural lecture at the University of Göttingen in 1854 entitled "On the Assumptions as the Basis of Geometry" is often considered the source of Riemann's geometry. In this talk, Riemann sees the surface itself as a separate geometric entity, rather than as a mere geometric entity in Euclidean space.

He first developed the concept of space, proposing that the object of geometry should be a multiplicity generalized quantity, and that the points in space can be n real numbers (x1,..., x) as coordinates. This is the original form of the modern n-dimensional differential manifold, which laid the foundation for describing natural phenomena in abstract space. This spatial geometry should be based on infinite proximity to two points (x1,...x) and (x1+dx1,..., x +dx ) ), measured by the positive definite quadratic form determined by the square of the length of the differential arc.

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Ellipsical geometry is Riemannian geometry. Geometry on Riemann manifolds. The German mathematician Riemann developed the theory of geometry in the mid-19th century.

Riemann's inaugural lecture at the University of Göttingen in 1854 entitled "On the Assumptions as the Basis of Geometry" is often considered the source of Riemann's empty potato geometry. In this talk, Riemann sees the surface itself as a separate geometric entity, rather than as a mere geometric entity in Euclidean space. He was the first to develop the concept of space, and proposed that the object of geometry should be a multiplicity of generalized quantities, and the points in space can be n real numbers (x1,..., x) as coordinates.

This is the original form of the modern n-dimensional differential manifold, which laid the foundation for describing natural phenomena in abstract space. This spatial geometry should be based on infinite proximity to two points (x1,...x) and (x1+dx1,..., x +dx ) ), measured by the positive definite quadratic form determined by the square of the length of the differential arc.

Riemann recognized that a metric is just a structure added to a manifold and that there can be many different measures on the same manifold. Mathematicians before Riemann only knew that there was an induced metric ds2=edu2+2fdudv+gdv2 on the surface s in three-dimensional Euclidean space e3, i.e., the first elementary form of the chain, but did not realize that s could also have a metric structure independent of the three-dimensional Euclidean geometry. Riemann realized the importance of distinguishing between induced metrics and independent Riemann metrics, and thus got rid of the shackles of induced metrics in classical differential geometry surface theory, founded Riemann geometry, and made outstanding contributions to the development of modern mathematics and physics.

Hope it helps!

Five points. The general equation for conic curves can be ascertained: ax +bxy+cy +dx+ey+f=0 (1).

When a,c are not all 0, and b -4ac < 0, it represents an ellipse.

Although there are 6 parameters in equation (1), since a,c are not all 0, you might as well let a≠0, so equation (1) can be reduced to.

x²+b'xy+c'y²+d'x+e'y+f'=0 (2)

In this way, five points are known to solve the system of five-element linear equations.

However, the condition for the only solution of equation (2) is that the coefficient determinant is not 0, and these five points still need certain constraints.

Generally, when these five points are connected to form a convex pentagon, the equation of the ellipse can be obtained.

Five points are required, the equation is (x-m) 2 a 2+(y-n) 2 b 2=1(a,b>0), first substitute four points to find a,b,m,n, but the solution m,n is not unique, need to substitute a point of verification, in order to choose.

Definition of an ellipse.

The sum of the distances from the two fixed points f1 and f2 in the plane is equal to the constant (greater than |f1f2|The trajectories of the points are called the ellipse, the two fixed points are called the focal points of the ellipse, and the distance between the two focal points is called the focal length. Second Definition:

An ellipse can be obtained by straightening the conical plane with a plane.

The standard equation for an ellipse.

The basic properties of the ellipse and related concepts.

The relative position of points and ellipses.

Tangents and normals of an ellipse.

The point is about the tangent chord and pole line of the ellipse.

The area of the ellipse.

I will set two points (x1, y1), (x2, y2), and the oblique argument rate is k1, k2 then k1k2=y1y2 difference change deficiency x1x2=-1 4

According to. x^2/16+y^2/4=1

y^2=4-x^2/4

So [sqrt(4-x1 2 4)*sqrt(4-x2 2 4)] x1x2=-1 4

It can be transformed. x2^2=16-x1^2

op|^2+|oq|^2=x1^2+y1^2+x2^2+y2^2=4-(3/4)*x1^2+4-(3/4)*x2^2

Because it's an isosceles triangle.

So |mf1|=8

mf2|=4

cosf1f2m=1/4

The ordinate of the point m is.

The root number 15 abscissa is.

It's the definition of an ellipse.

The sum of the distances from the plane to the fixed points f1 and f2 is equal to the constant (greater than |f1f2|) of the trajectory of the moving point p.

The formula for the distance between two points in the plane.

The square of the distance between the point m(m,n) and the point f1(-4,0).

m+4)²+n-0)²=8²

Apparently here PF2 and F1F2 are the waists of an isosceles triangle.

So pf2=f1f2=2c

The bottom angle is 60 degrees.

Because the outer angles of the triangle are equal to the sum of the two inner angles that are not adjacent to him.

Therefore, the angle between PF2 and the positive direction of the X axis is 60 degrees.

So the distance from F2 to X=3A2=Pf2*Cos60 degrees =C, and the distance from F2 to X=3A2 is 3A2-C

So c=3a2-c

So e=ca=3 4

Let p be in the first quadrant and the coordinates are (3a 2,n).

pf1f2=30°, pf1=2n, c+3a 2=root3n, so the solution is: pf1=2(c+3a 2) (root3)=(2root3)c 3+(root3)a

PF2=F1F2=2C, and PF1+PF2=2A (2 roots3)C 3+(Roots3)A+2C=2A, and the solution is E=C A=(9-5 roots3) 4