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Anonymous users2024-02-05Summary: There are two situations:
When the focus is on the x-axis, the standard equation for an ellipse is: x 2 a 2+y 2 b 2=1,(a>b>0);
When the focus is on the y-axis, the standard equation for an ellipse is: y2 a 2+x 2 b 2=1,(a>b>0);
where a2-c2=b2 (which is the key).
1. Elliptical focus.
When the focus is on the x-axis, the focus coordinates are f1(-c,0), f2(c,0).
When the focus is on the y-axis, the focal coordinates are f1(0,-c) and f2(0,c).
2. The geometric properties of the ellipse.
x,y.
When the focus is on the x-axis, -a x a, -b y b
When the focus is on the y-axis, -b x b, -a y a
c^2=a^2-b^2
3. Symmetry.
Regardless of whether the focus is on the x-axis or the y-axis, the ellipse is always symmetrical with respect to the x-y origin.
4. Vertex:
When the focus is on the x-axis: the vertices of the major axis: (-a,0),(a,0).
Minor axis vertices: (0,b), (0,-b).
When the focus is on the y-axis: the vertices of the major axis: (0,-a),(0,a).
Minor axis vertices: (b,0),(b,0).
Pay attention to which axis the long and short axes represent, which is easy to cause confusion here, and it is necessary to combine numbers and shapes to gradually understand thoroughly.
5. Equation derivation.
If the sum of the distances from one moving point to two fixed points in a plane is equal to the fixed length, then the trajectory of this moving point is called an ellipse.
Suppose (note that all assumptions are just for the sake of deriving elliptic equations) that the moving point is and the two fixed points are sum, then according to the definition, the trajectory equation of the moving point satisfies (definition):
Among them is the fixed length.
Using the formula for the distance between the two points, we can get :,, substituting it into the definition, and we get:
At that time, the juxtaposition could be further simplified:
Because, divide both sides of the equation together, and you get:
Then the equation is the trajectory equation of the moving point, that is, the equation of the ellipse. This form is also the standard equation for an ellipse.
If the image of an ellipse is represented in a Cartesian coordinate system, then the two fixed points in the above definition are defined on the x-axis. If the two fixed points are changed to the y-axis, the standard equation for another ellipse can be found in the same way:
In the equation, the set is called the major axis length, the minor axis length, and the fixed point is called the focal point, then it is called the focal length. In the process of hypothesis, it is assumed, and if you don't assume this, you will find that you can't get an ellipse. At that time, the trajectory of this moving point was a line segment; At that time, there was no actual trajectory at all, and at this time, its trajectory was called an imaginary ellipse.
Also note that in the hypothesis, there is one more place:
6. It is generally considered that the circle is a special case of the ellipse. (You must pay attention to the trade-offs when taking the exam).
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Anonymous users2024-02-04The standard equation for an ellipse is as follows:
When the focus is on the x-axis, the standard equation for an ellipse is: x 2 a 2+y 2 b 2=1,(a>b>0);
When the focus is on the y-axis, the standard equation for an ellipse is: y2 a 2+x 2 b 2=1,(a>b>0);
where a2-c 2=b 2.
Derivation: PF1+PF2>F1F2 (p is the point on the ellipse, f is the focal point).
Polar coordinate equations
One focal point is at the origin of the polar coordinate system, and the other is in the positive direction of 0=0) r=a(1-e2) (1-ecose) (e is the eccentricity of the ellipse = c a).
General equations
ax2+by2+cx+dy+e=0(a>0, b>0, and asubb).
Parametric equations
x=acose，y=bsine。
Common problems of ellipses and solutions
For example, if there is a cylinder that is cut off to a cross-section, it is proved below that it is an ellipse (with the first definition above): if you squeeze two hemispheres with the same radius as the radius of the cylinder from the ends of the cylinder to the middle, and they stop when they touch the cross-section, then you will get two common points, which are obviously the tangent points of the cross-section and the sphere.
Let the two points be f1 and f2 for any point p on the cross-section, pass p to make the bus bar q1 and q2 of the cylinder, and the large circle tangent to the sphere and the cylinder intersect q1 and q2 respectively pf1 and q2, so pf1 + pf2 = q1q2 is known by definition 1: the cross-section is an ellipse, and with f1 and f2 as the focus, the oblique section of the cone (not through the bottom surface) can also be proved to be an ellipse by the same method.
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Anonymous users2024-02-03The definition of an ellipse with the standard equation is as follows:
When the focus is on the x-axis, the standard equation for the ellipse is: x 2 a 2 + y 2 b 2 = 1, (a>b>0).
When the focus is on the y-axis, the standard equation for an ellipse is: y2 a 2+x 2 b 2=1,(a>b>0);
where a2-c 2=b 2.
Derivation: PF1+PF2>F1F2 (p is the point on the ellipse, f is the focal point).
Polar coordinate equations
One focal point is at the origin of the polar coordinate system, and the other is in the positive direction of 0=0) r=a(1-e2) (1-ecose) (e is the eccentricity of the ellipse = c a).
General equationsax2+by2+cx+dy+e=0(a>0, b>0, and asubb).
Parametric equationsx=acose，y=bsine。
Common problems of ellipses and solutions
For example, if there is a cylinder that is cut off to obtain a cross-section, it is proved below that it is an ellipse (using the first definition above): if you squeeze two hemispheres with a radius equal to the radius of the cylinder from the ends of the cylinder to the middle, and they stop when they touch the cross-section, then you will get two common points, which are obviously the tangent points between the lead cross-section and the ball.
Let the two points be f1 and f2 for any point p on the cross-section, the bus bar q1 and q2 of the cylinder through p, and the great circle tangent to the ball and the cylinder intersect q1 and q2 respectively pf1 = pq1 and pf2 = pq2, so pf1 + pf2 = q1q2 is known by definition 1: the cross-section is judged to be an ellipse, and with f1 and f2 as the focus, the oblique section of the cone (not through the bottom surface) can also be proved to be an ellipse by the same method.
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Anonymous users2024-02-02The 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|）。
The standard equation for an ellipse is divided into two cases:
When the focus is on the x-axis, the standard equation for an ellipse is: x 2 a 2+y 2 b 2=1,(a>b>0);
When the focus is on the y-axis, the standard equation for an ellipse is: y2 a 2+x 2 b 2=1,(a>b>0);
In mathematics, an ellipse is a curve in a plane around two focal points such that for each point on the curve, the sum of the distances to the two focal points is constant. 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.
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Anonymous users2024-02-011. The elliptic eccentricity is defined as the ratio of the focal length on the ellipse to the major axis, (range: 0e=c a(02c.) The greater the eccentricity, the flatter the ellipse; The smaller the eccentricity, the closer the ellipse is to the circle.
2. The focal distance of the ellipse: the distance between the focal point of the ellipse and its corresponding alignment (such as the focal point (c,0) and the alignment x= a 2 c) is a 2 c-c=b 2 c
3. Focus on the x-axis: |pf1|=a+ex |pf2|=a-ex(f1, f2 are the left and right focus, respectively).
4. The radius of the ellipse over the right Miga as the focal point r=a-ex.
5. The radius of the left focal point r=a+ex.
6. Focus on the y-axis: |pf1|=a+ey |pf2|=a-ey(f2,f1 are the upper and lower focus, respectively).
7. The diameter of the ellipse: the distance between the straight line perpendicular to the x-axis (or y-axis) and the two intersections of the ellipse, i.e., |ab|=2*b^2/a。
8. If the center is at the origin, but the position of the focus is not clear on the x-axis or y-axis, the equation can be set to beat the chain mx +ny =1(m>0, balance n>0, m≠n). That is, a unified form of the standard equation.
9. 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
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Anonymous users2024-01-313.Over the left focal point f(-1,0), the linear equation of the parallel training in the pat qin v=(1,1) is x+1=y, and the elliptic equation x 2 4+y 2 hits the middle of the bi 3=1, and obtains.
3x 2+4(x 2+2x+1)=12, 7x 2+8x-8=0,=64+4*7*8=8*36, let a(x1,y1),b(x2,y2), then |x1-x2|= 7, so |ab|=|x1-x2|√2=24/7.
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Anonymous users2024-01-30The ellipse and its standard equation are divided into two cases: when the focus is on the x-axis, the standard equation for the ellipse is: x 2 a 2+y 2 b 2=1, (a>b>0); When the focus is on the y-axis, the standard equation for an ellipse is:
y^2/a^2+x^2/b^2=1，(a>b>0)；where a2-c 2=b 2. Derivation: PF1+PF2>F1F2 (p is the point on the ellipse, f is the focal point).
Regardless of whether the focus is on the x-axis or the y-axis, the ellipse is always symmetrical with respect to the x-y origin. Vertex: When the focus is on the x-axis:
Axial vertices: (-a,0),(a,0); Minor axis vertices: (0,b), (0,-b).When the focus is on the y-axis:
Vertices of the major axis: (0,-a),(0,a); Minor axis vertices: (b,0),(b,0).
The elliptical mirror (a three-dimensional figure formed by rotating the ellipse 180 degrees on the long axis of the ellipse, and all its inner surfaces are made into reflective surfaces, hollow) can reflect all the light emitted from one focal point to another focal point;
Elliptical lenses (some of which are elliptical) have the effect of concentrating light (also called convex lenses), such as reading glasses, magnifying glasses, and farsighted glasses (these optical properties can be proved by the method of rebuttal). Eccentricity range: 0The smaller the eccentricity, the closer it is to the circle, and the larger the ellipse, the flatter the ellipse.
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