Conic curve is a difficult point in math exam, so what are the related knowledge points? The following summary of conic knowledge points is what I want to share with you. Welcome to browse.
Summary of knowledge points of conic curve Application of conic curve
Perspective of test sites
First, the examination outline
1. The objective function will be established according to the conditions, and the maximum value and range of variables will be studied, and the maximum value of some quantities will be found by using "combination of numbers and shapes" and "geometric method".
2. Further consolidate the method to solve related application problems with the definition and properties of conic curve.
Second, the placement of propositions
1. Investigate the application of conic equation in special background such as geographical location, and transform the highway engineering cost problem into a mathematical model to solve the maximum distance problem, such as1;
2. Examine the basic knowledge such as straight line and parabola, and examine the ability to solve problems by analytic geometric analysis, as shown in Example 2;
3. Examine the concept and equation of hyperbola, and examine the examinee's ability to analyze and solve practical problems, as shown in Example 3.
Detailed analysis of typical cases
Example 1: (2004? Fujian) As shown in the figure, plot B is 4km east of plot A, and plot C is 2km east of 300 north of plot B. The distance from any point on the PQ (curve) along the river to A is 2km longer than that to B. Now, it is necessary to choose a location M on the curve PQ to build a wharf and transport the goods to B and C. According to the calculation, the road construction costs from M to B and M to C are A respectively.
A.(2-2)a ten thousand yuan B.5a ten thousand yuan
C.(2+ 1) Ten thousand yuan D.(2+3) Ten thousand yuan.
Analysis: let the total cost be y ten thousand yuan, then y=a? MB+2a？ host
The distance from any point on PQ (curve) along the river to A is 2km farther than that to B,
? Curve PG is a hyperbola, b is the focus, a= 1, c=2.
Let m be perpendicular to the directrix L corresponding to the hyperbola focus B, and let the vertical foot be d (as shown in the figure). From the second definition of hyperbola, it is =e, which means MB=2MD.
? y= a？ 2MD+ 2a？ MC=2a？ (MD+MC)？ 2a？ CE。 (where CE is the vertical section from point c to directrix l).
∫CE = g b+ BH =(c-)+BC？ cos600=(2-)+2？ =.？ y？ 5a (ten thousand yuan).
Answer: B.
Example 2: (In 2004? Beijing, Li 17) as shown in the figure, through the parabola y2 = 2px (p >; 0) the last fixed point p (x0, y0) (y0 >; 0), make two straight lines and intersect parabolas at A(x 1, y 1) and b (x2, y2) respectively.
(1) Find the distance from a point with ordinate on a parabola to its focus f;
(2) When the slopes of PA and PB exist and the inclination angles are complementary,
It is proved that the slope of straight line AB is a non-zero constant.
Analysis: (1) When y=, x=.
The directrix equation of parabola y2=2px is x=-, which is defined by parabola.
The distance sought is.
(2) Let the slope of the straight line PA be kPA and the slope of the straight line PB be kPB.
From y 12=2px 1 and y02=2px0, the subtraction is obtained.
Therefore, for the same reason,
According to the complementary tilt angles of PA and PB, that is,
So, therefore.
Let the slope of the straight line AB be kAB, and subtract 0, 0 to get it, so. Will be replaced by,
So kAB is a non-zero constant.
Example 3: (In 2004? Guangdong) A center received reports from three observation points due east, due west and due north: two observation points due west and due north heard a loud noise at the same time, and the time at the due east observation point was 4s later than the other two observation points. Knowing that the distance from each observation point to the center is 1020m, try to determine the location of the loud noise. (Suppose that the speed of sound propagation was 340m/s at that time, and all relevant points were in the same place.
Analysis: As shown in the figure, a rectangular coordinate system is established with the newspaper receiving center as the origin O, the due east and north directions as the X axis and the Y axis forward. Let A, B and C be observation points in the west, east and north respectively, then A (-1020,0), B (1020,0), C (.
Let P(x, y) be the occurrence point of loud noise, and A and C hear loud noise at the same time, and get |PA|=|PC|.
So P is on the middle vertical line PO of AC, and the equation of PO is Y =-X. Because point B heard the explosion 4s later than point A, |PB|-|PA|=340? 4= 1360.
Point p is defined by hyperbola as being on a hyperbola focusing on a and b,
According to the meaning, a=680, c= 1020, b2=c2-a2= 10202-6802=5? 3402,
Therefore, the hyperbolic equation is. Substitute y=-x into the above formula to get x=? 680,
∵| PB | >； |PA|，？ X=-680, y=680, which is P (-680,680), so PO=680.
A: The loud noise occurred 450 meters northwest of the receiving center and 680 meters away from the center.
Common misunderstandings
The practical application of 1. conic curve has a certain real life background, and candidates have different degrees of difficulty in mathematical modeling and solving. Returning to the definition, linking practical problems with them and flexible transformation are the key to solve such problems;
2. The problem of fixed point, quantification and fixed value of conic curve is a fixed property hidden in the curve equation. Candidates can only analyze the superficial problems, but can't summarize their substantive conclusions, which makes the research on the problems linger. We should pay attention to the gradual induction from the special to the general, and try to deduce and demonstrate this kind of problem.
Basic exercises
1.(2005? Chongqing) If the moving point () changes on the curve, the maximum value of is () a.b. 。
C.D.2
2.(2002? Country), then the eccentricity range of conic curve is () a.b.c.d
3.(2004? The axial section of the wine glass is a part of parabola.
The equation of is x2=2y, y? [0, 10] Put a cleaning ball in the cup, and it is required that the cleaning ball can
Wipe the bottom of the glass (as shown in the figure), then the maximum radius of the cleaning ball is ()
A.B. 1
4.(2004? There is a point P on the ellipse of Taizhou Sanmiao, where F 1 and F2 are the left and right focal points of the ellipse, and △F 1PF2 is a right triangle, then such a point P has ().
A.2 B.4 C.6 D.8
5.(2004? Let f be the right focus of the ellipse, and at least 2 1 different points Pi(i= 1, 2,3, ...) are on the ellipse, so that |FP 1|, |FP2|, |FP3|, ... form a arithmetic progression with a tolerance of d, then
6.(2004? The two chapters of the textbook "straight line on coordinate plane" and "conic curve" reflect the essence of analytic geometry.
7.(2004? Zhejiang) It is known that the center of hyperbola is at the origin,
The right vertex is A( 1, 0), and points P and Q are on the right branch of hyperbola.
The distance from point M(m, 0) to straight line AP is 1.
(1) If the slope of the straight line AP is k and |k|? [],
The range of real number m;
(2) When m=+ 1, the heart of △APQ is exactly the point m,
Find the equation of this hyperbola.
8.(2004? As shown in the figure, the straight line y=x and parabola.
The straight line y=x2-4 passes through point A and point B, and the vertical plane of line segment AB is flat.
The bisector and the straight line y=-5 intersect at the q point.
(1) Find the coordinates of point Q;
(2) When p is a parabola, it is located below the line segment AB.
When moving points (including a and b), what? Maximum value of OPQ area.
9.(2004? At 9: 00 10 on June 5th, 2003, Shenzhou 5 manned spacecraft was launched. At 9: 09: 50, it accurately entered the scheduled orbit and began to survey the sky. The orbit is an ellipse with the center of the earth as the focus. Select the coordinate system as shown. The center of the ellipse is at the origin. Perigee A is 200km from the ground, and apogee B is 350km from the ground.
(1) Find the spacecraft elliptical orbit equation;
(2) After the spacecraft circled the earth 14 times, at 5: 59 on June 16, the return module was separated from the propulsion module, ending the sky survey flight. The spacecraft flew about * * * in the sky and asked the spacecraft to patrol.
What is the average speed of flying every day? The result is accurate.
To 1 km/s) (Note: km/s means km/s)
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