Definition of 1. ellipse: In the definition of ellipse, the condition that the sum of the distances between a moving point and two fixed points F 1 and F2 in the plane is greater than |F 1F2| cannot be ignored. If the sum of these distances is less than | F 1F2|, then such a point does not exist; If the sum of distances is equal to
| F 1F2|, then the trajectory of the moving point is the line segment F 1F2.
2. The standard equation of ellipse: x? /a？ +y？ /b？ = 1(a>b>0)，y？ /a？ +x？ /b？ = 1(a>b>0)。
3. The standard equation discrimination method of ellipse: Which axis is the focus of discrimination? Just look at the size of the denominator: if x? The denominator of the item is greater than y? Denominator of the term, the focus of the ellipse is on the X axis, otherwise the focus is on the Y axis.
4. The method of solving the elliptic standard equation: (1) correctly judge the focus position; ⑵ After setting the standard equation, use the undetermined coefficient method to solve it.
(2) Simple geometric properties of ellipse
1. Geometric properties of ellipse: Let the elliptic equation be x? /a？ +y？ /b？ = 1(a>b>0)。
(1) range: -a≤x≤a, -b≤x≤b, so the ellipse is located in the rectangle surrounded by straight lines X = A and Y = B (2) Symmetry: symmetrical about the x axis and y axis respectively, and symmetrical about the center of the origin. The center of symmetry of an ellipse is called the center of the ellipse.
⑶ Vertex: There are four A 1(-a, 0), A2(a, 0)B 1(0, -b), B2(0, b).
The line segments A 1A2 and B 1B2 are called the major axis and minor axis of the ellipse respectively. Their lengths are equal to 2a and 2B respectively, and a and b are called the major axis and minor axis of the ellipse respectively. Therefore, there are four intersections between an ellipse and its axis of symmetry, which are called the vertices of the ellipse.
⑷ Eccentricity: The ratio of the focal length of an ellipse to the length of its major axis, e=c/a, is called eccentricity of an ellipse. Its value indicates the flatness of the ellipse. The closer to E < 1. E to 1, the flatter the ellipse becomes. On the contrary, the closer E is to 0, the closer an ellipse is to a circle.
2. The second definition of ellipse
⑴ Definition: When the ratio of the distance between a moving point M and a vertex on a plane and its distance to a fixed straight line is constant E = C/A (E < 1), the trajectory of this moving point is an ellipse.
(2) directrix: According to the symmetry of ellipse, x? /a？ +y？ /b？ There are two directrix = 1 (a > b > 0), and their equation is x = (a? /c)。 For ellipse y? /a？ +x？ /b？ = 1 (A > B > 0), just change x into y, that is, y=
(a？ /c)。
3. The focal radius of the ellipse: the line segment connecting any point on the ellipse with its focal point is called the focal radius of the point.
Let f 1 (-c, 0) and F2 (c, 0) be ellipses x? /a？ +y？ /b？ = 1 (a > b > 0) and M(x, y) is any point on the ellipse, then the radius length of the two focal points is |MF 1|=a+ex, |MF2|=a+ex.
When the focal radius is involved in an ellipse, it is often easier to solve the problem by using the knowledge of focal radius.
Of the four main elements of an ellipse, A, B, C and E, A? =b？ +c？ E=c/a, so only two independent conditions are needed to determine the standard equation of ellipse.
4. Parametric equation of ellipse
Ellipse x? /a？ +y？ /b？ The parameter equation of = 1 (a > b > 0) is x = acos θ and y = bsin θ (θ is the parameter).
Note: (1) The parameter θ here is called the eccentricity angle of ellipse. The eccentricity angle θ of the point P on the ellipse is different from the inclination angle α of the straight line OP: tan α = (b/a) tan θ;
⑵ The parametric equation of ellipse can be expressed by equation X? /a？ +y？ /b？ = 1 and trigonometric identity sin? θ+cos？ θ= 1, so the essence of the parametric equation of ellipse is triangular substitution.
5. Inside and outside the ellipse
(1) Point P(x0, y0) is within ellipse X? /a？ +y？ /b？ = 1 (a > b > 0)，x0？ /a？ +y0？ /b？ < 1.
(2) Is point P(x0, y0) in ellipse X? /a？ +y？ /b？ = 1 (a > b > 0)，x0？ /a？ +y0？ /b？ > 1.
6. The tangent equation of ellipse
(1) ellipse x? /a？ +y？ /b？ = 1 (a > b > 0) The tangent equation at point P(x0, y0) is (x0? x)/a？ +(y0？ y)/b？ = 1.
(2) through the ellipse x? /a？ +y？ /b？ = 1 (a > b > 0) The tangent chord equation of two tangents cited by a point P(x0, y0) is (x0? x)/a？ +(y0？ y)/b？ = 1.
(3) ellipse x? /a？ +y？ /b？ = 1 (a > b > 0) tangent to the straight line Ax+By+C=0 if a? Answer? +B？ b？ =c？
(3) hyperbola and its standard equation
Definition of 1. hyperbola: The locus of moving point m with absolute value equal to constant 2a (less than |F 1F2|) on the plane is called hyperbola. In this definition, we should pay attention to condition 2a.
If 2a=|F 1F2|, the trajectory of the moving point is two rays; If 2a > | f 1f2 |, there is no trajectory. If | mf 1 | < | mf2 |, the trajectory of moving point m is only a branch of hyperbola; if | mf 1 | > | mf2 |, the trajectory is
The other branch of the hyperbola, which consists of two branches, should be defined as "the absolute value of the difference".
2. The standard equation of hyperbola: x? /a？ -Really? /b？ = 1 and y? /a？ +x？ /b？ = 1 (a > 0，b > 0)。 Here b? =c？ -a？ , where |F 1F2|=2c. Pay attention to the similarities and differences between a, b and c here and their relationship with the ellipse.
3. the standard equation of hyperbola is: if x? If the coefficient of this term is positive, the focus is on the X axis; If the coefficient of this term is positive, the focus is on the Y axis. For hyperbola, a is not necessarily greater than b, so you can't compare denominators like an ellipse.
Small to determine which axis the focus is on.
4. To solve the standard equation of hyperbola, we should pay attention to two problems: correctly judge the position of focus; ⑵ After setting the standard equation, use the undetermined coefficient method to solve it.
(D) Simple geometric properties of hyperbola
1. hyperbola: x? /a？ -Really? /b？ The real axis length of = 1 is 2a, the imaginary axis length is 2b, and the eccentricity e = c/a > 1. The greater the eccentricity e, the larger the opening of hyperbola.
2. hyperbola: x? /a？ -Really? /b？ Is the asymptote equation of = 1 y = (b/a) or expressed as: x? /a？ -Really? /b？ =0. If the asymptote equation of hyperbola is known as y = (m/n) x, that is, MX ny = 0, then the equation of hyperbola has the following form: m? x？ -
n？ y？ =k, where k is a nonzero constant.
3. The second definition of hyperbola: The locus of a point whose distance from a fixed point (focus) to a fixed line (directrix) on the plane is a constant (eccentricity) greater than 1 is called hyperbola. For hyperbola: x? /a？ -Really? /b？ = 1, and its focal coordinate is (-c, 0).
And (c, 0), their corresponding directrix equation is x=-a? /c and x=a? /C. hyperbola: x? /a？ -Really? /b？ The focal radius formula of = 1 (a > 0, b > 0) |PF 1|=|e(x+a? /c)|，|PF2|=|e(-x+a？ /c)|。
4. The interior and exterior of hyperbola
(1) point P(x0, y0) is in hyperbola x? /a？ -Really? /b？ = 1 (a > 0，b > 0)，x0？ /a？ -What about you? /b？ < 1.
(2) Is point P(x0, y0) in hyperbola X? /a？ -Really? /b？ = 1 (a > 0，b > 0)，x0？ /a？ -What about you? /b？ > 1.
5. The relationship between hyperbolic equation and asymptote equation.
(1) If the hyperbolic equation is x? /a？ -Really? /b？ = 1 gives asymptote equation: x? /a？ y？ /b？ =0 gives y = (a/b) x.
(2) If asymptote equation is y = (a/b) x, x? /a？ y？ /b？ =0, hyperbola can be set to x? /a？ -Really? /b？ =λ.
(3) If hyperbola and x? /a？ -Really? /b？ = 1 has a common asymptote, which can be set as x? /a？ -Really? /b？ = λ (λ > 0, focusing on X axis, λ < 0, focusing on Y axis).
6. Tangent equation of hyperbola
(1) hyperbolic x? /a？ -Really? /b？ = 1 (a > 0, b > 0) The tangent equation at point P(x0, y0) is (x0? x)/a？ -(y0？ y)/b？ = 1.
(2) hyperbolic x? /a？ -Really? /b？ = 1 (a > 0, b > 0) The tangent chord equation of two tangents cited by a point P(x0, y0) is (x0? x)/a？ +(y0？ y)/b？ = 1.
(3) hyperbolic x? /a？ -Really? /b？ = 1 (a > 0, b > 0) tangent to the straight line Ax+By+C=0 if a? Answer? -B? b？ =c？ .
(5) Standard equation and geometric properties of parabola
1. Definition of parabola: The trajectory of a point on a plane whose distance is equal to the distance of an alignment (L) is called parabola. This fixed point F is called the focus of parabola, and this fixed line L is called the directrix of parabola.
It should be emphasized that point F is not on the straight line L, otherwise the trajectory is a straight line passing through point F and perpendicular to L, not a parabola.
2. There are four types of parabolic equations:
y？ =2px、y？ =-2px、x？ =2py、x？ =-2py。
For the above four equations, we should pay attention to their laws: which axis is the symmetry axis of the curve, and the terms in the equation are linear terms; There is a plus sign in front of the first term, and the opening direction of the curve is in the positive direction of X axis or Y axis; The first term is preceded by the opening of the negative curve.
The direction of the mouth is opposite to the x axis or the y axis.
3. Geometric properties of parabola, taking standard equation y2=2px as an example.
(1) range: x ≥ 0;
(2) Symmetry axis: the symmetry axis is y=0, which can be seen from the equation and the image;
(3) Vertex: O (0,0), Note: Parabola is also called centerless conic curve (because there is no center);
(4) Eccentricity: e= 1, because E is a constant, the shape change of parabola is determined by P in the equation;
(5) the alignment equation x =-p/2;
(6) formula of focal radius: a point on the parabola is p (x 1, y 1), and f is the focus of the parabola. For four kinds of parabolas, the focal radius formula is (p > 0):
y？ =2px，| PF | = x 1+p/2； y？ =-2px，|PF|=-x 1+p/2
x？ =2py，| PF | = y 1+p/2； x？ =-2py，|PF|=-y 1+p/2
(7) Focal chord length formula: For the chord length passing through the parabolic focal point, the focal radius formula can be derived. Let the chord passing through the focus f of the parabola y2 = 2px (p > o) be AB, A (x 1, y 1), B (x2, y2), and the inclination angle of AB be α, then ① |
AB | = X+X+P2Ab | = 2p/(Sina)? These two formulas are only suitable for solving the chord length of the focus, and only the chord length formula can be used for other chords.
(8) Relationship between straight line and parabola: After the combination of straight line and parabola equation, a quadratic equation with one variable is obtained: x +bx+c=0. When a≠0, the judgment of the positional relationship between them is the same as that of ellipse and hyperbola, so the judgment method can be used; But if a=0, the straight line is a throw.
The symmetry axis of an object line or a straight line parallel to the symmetry axis. In this case, the straight line intersects the parabola, but there is only one common point.
4. parabola y? The fixed point on =2px can be set to P(y0? /2p, y0) or P(y0? /2p, y0) or P(x0, y0), where y0? =2px0。
5. Quadratic function y=ax? +bx+c=a(x+b/2a)？ + [ (4ac-b？ The image of /4a ](a≠0) is a parabola: (1) The vertex coordinates are [-b/2a, (4ac-b? )/4a]； (2) The coordinate of the focal point is [-b/2a, (4ac-b? + 1)/4a]； (3) Alignment
Cheng is y=(4ac-b? + 1)/4a。
6. Inside and outside parabola
(1) point P(x0, y0) is in parabola y? = 2px (p > 0)，y？ < 2px(p>0)。
Is the point P(x0, y0) on the parabola y? = 2px (p > 0)，y？ > 2px(p>0)。
(2) Is the point P(x0, y0) in the parabola y? =-2px (p > 0)，y？ 0)。
Is the point P(x0, y0) on the parabola y? =-2px (p > 0)，y？ >-2px(p>0)。
(3) Is the point P(x0, y0) in the parabola X? = 2py (p > 0)，x？ < 2py(p>0)。
Is the point P(x0, y0) in the parabola X? = 2py (p > 0)，x？ > 2py(p>0)。
(4) Is the point P(x0, y0) in the parabola X? =-2py (p > 0)，x？ 0)。
Is the point P(x0, y0) in the parabola X? =-2py (p > 0)，x？ >-2py(p>0)。
7. Tangent equation of parabola
(1) parabola y? The tangent equation of a point P(x0, y0) on = 2px (p > 0) is y0? y=p(x+x0)。
(2) through the parabola y? = 2px (p > 0), and the tangent chord equation of two tangents cited by a point P(x0, y0) is y0? y=p(x+x0)。
(3) parabola y? The condition that = 2px (p > 0) is tangent to the straight line Ax+By+C=0 is pB? =2AC。
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