The elliptic properties prove that the ellipse has all the properties

The proportional formula is wrong, it should be af:fp=at:pp'=am':m'p (t is the intersection of the line and the x-axis).

pp'⊥m't,at⊥m't

pp'm'∽△atm'AT:PP is available'=am':m'p………1) It is also defined by the second ellipse: the ratio of the distance from a point on the ellipse to a certain focal point and the distance from the point to the corresponding alignment of the focal point is the eccentricity.

pf:pp'=af:at=e

af:fp=at:pp'………2)

Combining (1) and (2) gives af:fp=at:pp'=am':m'p and then according to the inverse theorem of the bisector theorem of the outer angle of the triangle is obtained m'Divide the PFT equally

Since this question is asked, there should be a certain foundation.

The optical properties of an ellipse are that light rays incident from one focal point and are reflected through the elliptical boundary to reach the other focal point.

Proof idea: Establish a coordinate system, set up a straight line equation (1) past the left focus, find the intersection point with the ellipse, and then find the tangent equation (2) of the derivation point, find the symmetrical linear equation about (3), and know (3) through the right focus, and prove it.

The projection of the left focal point F1 on the straight line PT is H, and the extension of F1H intersects F2P at the point Q, which can prove that the PT bisects the line segment F1Q perpendicularly, so that Qp=F1P, F1H=Hq, according to the ellipse definition, PF1 PF2=2A, and QP PF2=PF1 PF2=2A, that is, QF2=2A, since Ho is the median line of the triangle QF1F2, then HO=(1 2)QF2=A, thus proving your problem.

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

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

3. Eccentricity: e= (1-b 2 a).

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).

7. p is a point on the ellipse, a-c pf1 (or pf2) a+c.

8. The circumference of an ellipse is equal to the length of a particular sinusoidal curve in a period.

Focal radius

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。

1 Simple geometric properties of an ellipse.

Take the equation as an example:

1) Range: From the equation |x|≤a，|y|b, so the ellipse is located in a rectangle surrounded by the straight lines x= a, y= b.

2) Symmetry: The ellipse is both axisymmetric and centrally symmetrical, it has two axes of symmetry, a center of symmetry, generally for the curve f(x,y)=0, if the equation is unchanged with y, the curve is symmetrical with respect to the x-axis, and if the equation is unchanged with x, the curve is symmetrical with respect to the y-axis; If x is used to replace x and y is replaced by y, then the symmetry of the curve with respect to the origin should be understood and memorized in conjunction with the coordinates of the symmetry points of the x-axis, y-axis, and origin of the points p(x,y) respectively.

3) Vertices: There are four in total, ie.

They are the intersection points of the ellipse and the coordinate axes, and when drawing an ellipse, you can draw these four vertices first, and then draw the general shape of the ellipse.

4) Eccentricity:

In the ellipse, a>c>0, 0 if a is unchanged

It is easy to see that the larger the e, the smaller the b, and the flatter the ellipse; The smaller the e, the larger the b, and the more round the ellipse, therefore, the eccentricity reflects how flat the ellipse is.

2 Second definition of ellipse.

The ellipse can also be seen as the ratio of the distance from the moving point to the fixed point f and to the fixed line 1 is equal to the constant e (0 From the symmetry, the ellipse has two quasi-lines, for the ellipse.

The alignment equation corresponding to f(c,0) is.

The alignment equation corresponding to f(c,0) is.

If the elliptic equation is.

Then the two alignment equations are.

From the second definition, if m is any point on the ellipse, the line 1 is the alignment corresponding to the focal point f, and the distance from m to 1 is d, then |mf|=ed, using this relation, the distance from a point on the ellipse to the focal point can be converted into the distance from it to the corresponding alignment, simplifying the operation.

3 Parametric equations for ellipses.

From elliptic equations.

Lenovo trigonometric formula, Ruoling.

That is, this is the parametric equation of the ellipse.

It indirectly reflects the relationship between two coordinates of one point p(x,y) on the ellipse.

When using the parametric equation of the ellipse to study the problem of maximum value, it is not necessary to eliminate the element through the ordinary equation, but to directly establish the univariate objective function about the angular parameter.

Apollonius's eight-volume Conic Theory of Curves (Conics) was the first to put forward the terms related to conic crossings, such as ellipse, parabola, and hyperbola, which are familiar today, and can be said to be the best work of ancient Greek geometry.

Until. Ten. At the turn of the sixth and seventeenth centuries, Kepler's discovery of the three laws of planetary motion made it clear that the orbit of a planet around the sun is an ellipse with the sun as its focus.

An ellipse is an extension of the definition of a circle, and it is a graph of all the points in a plane whose distance to two points is the sum of a fixed value.

These two points are called focal points, and the distance between two points is called focal length. When two focal points coincide, the ellipse becomes a circle.

The ellipse is a plane figure, and we usually find a way to establish a coordinate system to represent the plane shape. The position and orientation of the two focal points are not specified in the ellipse definition, so the size, position, and orientation of the ellipse can all be varied.

Since the ellipse is a symmetrical figure, satisfying both axis symmetry and center symmetry, therefore, for the sake of simplicity, the origin is usually selected as the symmetry center of the ellipse, and the two coordinate axes are used as the symmetry axes of the ellipse, then the focus of the ellipse should be two symmetrical points on the coordinate axis.

The elliptic property is the sum of the distance between the two fixed points f1 and f2 in the plane and the trajectory of the moving point p with a constant 2a is called an ellipse, where 2a >|f1f2|。This is the standard definition in the textbook and will not be described in detail. Any tangent of the ellipse is equal to the angle of the two focal radii at the tangent.

An ellipse is a closed conical section: a planar curve that is intersected by a cone and a plane. The ellipse has many similarities with the other two forms of conical sections:

Parabola and hyperbola, both are open and unbounded. The cross-section of a cylinder is elliptical unless the section is parallel to the axis of the cylinder.

Ellipses have a good optical property: light emitted from one focal point converges to the other. The proof of this miraculous nature is often explained by analytic geometry. Here is a simple method that can be explained by only geometric methods.

Let's describe the problem first: we know that the semi-major axis of the ellipse is A, the focus is F1F1 and F2F2, choose any point C on the ellipse (the collinear situation is easy to say, here you might as well think that C is not collinear with F1F1 and F2F2), make the angular bisector of C ll, and make the perpendicular line M of ll through point C, then M is the tangent of the ellipse.

This is somewhat similar to a high school question: we know that there are two villages F1, F2 and River M, and a pumping station p is to be built on M, and ask P in ** to make Pf1+Pf2Pf1+Pf2 the smallest. Be inspired, as evidenced below.

Proof idea: Add auxiliary lines - as CF1CF1 symmetrical line segment CA with respect to m. It is easy to prove that a, c, f2f2 are collinear. This is very similar to the pumping station problem: if you take the point p on m that is not c, then.

pa+pf2>ca+cf2=2a

pa+pf2>ca+cf2=2a

That is, pf1+pf2pf1+pf2 should also be greater than 2a, that is, the p point should fall outside the ellipse. This means that the line m has only one intersection point with the ellipse. i.e. m is the tangent of the ellipse.

Ellipse properties: The range and symmetry of the ellipse: (a b 0) in -a tantan x a, -b y b, the center of symmetry is the origin, and the axis of symmetry is the coordinate axis.

Vertices: a(a,0), b(-a,0), c(0,b), and d(0,-b).

Axis: Axis of symmetry: X-axis, Y-axis; The long axis is long|ab|=2a, the length of the short axis |cd|=2b, a is the length of the major semi-axis, and b is the length of the minor semi-axis.

Eccentricity range: 0The smaller the eccentricity, the closer it is to the circle, and the larger the ellipse, the flatter the ellipse.

The standard equation for an ellipse.

There are two standard equations for an ellipse, depending on the axis on which the focal point is located:

1. When the focus is on the x-axis, the standard equation is:

2. When the focus is on the y-axis, the standard equation is:

The sum of the distances from any point on the ellipse to f1 and f2 is 2a, and the distance between f1 and f2 is 2c. And b in the formula = a -c. b is a parameter set for the writing side.

Also: if the center is at the origin, but the position of the coke bucket point is not clear on the x-axis or y-axis, the equation can be set to mx +ny =1(m>0,n>0,m≠n). That is, a unified form of the standard equation.

The area of the ellipse is ab. An ellipse can be seen as a stretch of a circle in a certain direction, and its parametric equation is: x=acos, y=bsin

The tangent of the standard form of ellipse at the point (x0,y0) is: xx0 a +yy0 b =1. The slope of the elliptic tangent is: -b x0 a y0, which can be calculated by complex algebra.