椭圆的第一定义（First definition of ellipse）

椭圆的第一定义（First definition of ellipse） 椭圆的第一定义(First definition of ellipse) Data worth having From the usual study, accumulation and summary Where there is a problem, there must be some Please also criticize and correct me! First definition of ellipse I Tu Yu n The distance between the plane and the two point F and F'is equal to the fixed point 2A (2a>|FF'|). The locus of P is called an ellipse Namely: / PF / + / PF'/ =2a Two of them, F and F', are called the focus of the ellipse Two / FF', called the focal distance of ellipse focal length The second definition of ellipse The ratio of the distance between the plane to the point F and the fixed line is constant e (i. e. the eccentricity of the ellipse) The set of points (e=c/a) (fixed point F is not on the fixed line The constant is a positive number less than 1.) Point F is the focus of the ellipse A fixed line called elliptical directrix (the linear equation is x= + a^2/c or y= + a^2/c) According to the definition of other ellipse is an important character of ellipse is the slope of points on the ellipse and the short axis of the ellipse at the ends of the product line is a fixed value can be obtained: the slope plane and two fixed line product is constant K point orbit is elliptical At this point, K should meet certain conditions That is, the excluded slope does not exist Geometric properties of tangents and normals Theorem 1: let F1 and F2 be the two foci of elliptic C P is any point on the C If the line AB, cut the ellipse C at the point P Is / APF1= / BPF2 Theorem 2: let F1 and F2 be the two foci of elliptic C P is any point on the C If the line is C AB in normal p The AB / F1PF2. References to the above two theorems can be viewed by reference materials Computer graphics constraint The ellipse must have a diameter parallel to the X axis The other is parallel in diameter and Y axis Geometric ellipses that do not satisfy this condition are considered as closed curves in computer graphics Standard equation The high school textbook is in the plane Cartesian coordinate system The ellipse is described by an equation The "standard" in the standard equation of the ellipse refers to the center at the origin The axis of symmetry is the coordinate axis There are two standard equations for an ellipse Depending on the axis of the focus: 1) focus on the X axis The standard equation is: x^2/a^2+y^2/b^2=1 (a>b>0) 2) focus on the Y axis The standard equation is: x^2/b^2+y^2/a^2=1 (a>b>0) Among them, a>0 B>0 The larger half of a and B is the long axis of ellipse The shorter is the shorter half axis (the ellipse has two symmetrical axes Symmetric F points on the Y axis The shaft is cut by an ellipse There are two line segments Half of them are called the long semi axis of the ellipse and the short half axis or the half long axis and half short axis, when a>b The focus is on the X axis The focal length is 2* (a^2-b^2) ^0.5 The focal length and short half axis relationship: b^2=a^2-c^2, x=a^2/c and x=-a^2/c line equation C is the half focal length of an ellipse PS: if the center at the origin. But the focus is not clear when it is on the X axis or Y axis The equation can be set to mx^2+ny^2=1 (m>0 N>0 M = n) Uniform form of standard equation The area of an ellipse is pi ab An ellipse can be considered as a stretch of a circle in a certain direction Its parameter equation is: x=acos theta Y=bsin theta The standard form of ellipse is (x0 Y0) the tangent of the point is: xx0/a^2+yy0/b^2=1 LK general equation Ax^2; +Bxy+Cy^2; +Dx+Ey+F=0 (A.C is not 0) formula Area formula of an ellipse S= PI (PI) * a * B (a, B respectively is elliptical semimajor axis, short half axis length). S= or PI (PI) * A * B/4 (where A, B are the major axis of the ellipse, long short axis). Perimeter formula of ellipse Elliptic perimeter has no formula Having an integral or an infinite term expansion The precise calculation of the ellipse's circumference (L) uses the summation of integral or infinite series as L = formula [0, PI /2]4a * sqrt (1- (e*cost) 2) DT = 2 pi V ((a2+b2) /2) [] ellipse perimeter, where a is the elliptical semimajor axis, eccentricity e The definition of ellipticity of ellipse point to a focal distance and the point to the line corresponding to the ratio of the distance of focus Set the point P on the ellipse to a focal point, the distance is PF To the corresponding alignment distance of PL be E=PF/PL Alignment equation of ellipse X= + a^2/c Eccentricity formula of ellipse E=c/a (e<1, because of 2a>2c) The quasi elliptic focal distance: the focus of an ellipse and its corresponding directrix (such as focus (C, 0) and line x=+a2/C) distance value =b2/c Elliptic focusing radius formula |PF1|=a+ex0 |PF2|=a-ex0 The radius of the ellipse over the right focus r=a-ex The radius of the left focus r=a+ex Elliptical path: the distance between the straight line perpendicular to the X axis (or the Y axis) and the two intersection of the ellipse, A, B Numerical value =2b^2/a Relation between point and ellipse Point M (x0 Y0) elliptic x^2/a^2+y^2/b^2=1 Point in circle: x0^2/a^2+y0^2/b^2<1 Point on the circle: x0^2/a^2+y0^2/b^2=1 Point outside circle: x0^2/a^2+y0^2/b^2>1 Positional relation between line and ellipse Y=kx+m X^2/a^2+y^2/b^2=1 II X^2/a^2+ (kx+m) ^2/b^2=1 can be introduced Tangent Delta =0 From <0 without intersection The intersection of delta >0 can use chord formula: A (x1, Y1) B (X2, Y2) |AB|=d = V (1+k^2) |x1-x2| (1+k^2) = V (x1-x2) = ^2 (1+1/k^2 = |y1-y2|) tick tick (1+1/k^2) (y1-y2) ^2 Slope formula of ellipse X^2/a^2+y^2/b^2=1 over the ellipse (x) The tangent slope of Y is - (b^2) X/ (a^2) y Elliptic focusing triangle area formula If the angle F1PF2= S=b^2tan /2 theta theta. Application of elliptic parameter equation When the point on the ellipse is determined from the point or the distance from the fixed line The problem can be transformed into trigonometric function problem by parametric coordinates X=a * cos beta Y=b x sin beta A is half of the long axis Correlation property The figure obtained by a plane frustum of a cone (or column) may be elliptical It belongs to a conic section For example: there is a cylinder Be cut off to obtain a cross section The following proves that it is an ellipse (with the first definition above): A hemisphere with two radii equal to the radius of the cylinder is squeezed from the two ends of the cylinder to the center They stop when they hit the cross section Then you'll get two public points Obviously they are the cutting point of the cross section and the ball Set two points for F1, F2 For any point on the cross section, P P over the column to do the bus Q1, Q2 The great circle tangent to the ball and cylinder is given to Q1 and Q2 respectively Then PF1=PQ1, PF2=PQ2 So PF1+PF2=Q1Q2 By definition 1: cross section is an ellipse And take F1, F2 as the focal point after the same method It can also be proved that the oblique section of the cone (not through the bottom surface) is an ellipse Example: C:x^2/a^2+y^2/b^2=1 (a>b>0) known elliptical eccentricity for root 6/3 A short axis endpoint to the right focal distance of 3. V (1) find the equation of elliptic C (2) the line l:y=x+1 and the ellipse are given to A B two points P is one point on the ellipse The maximum value for PAB area. (3) in (2) for AOB on the basis of the area. First, analyze the distance from the short axis to the left and right focus, and to 2A The distance from the endpoint to the left and right focus is equal (ellipse definition) The appropriate 3 a= C/a= and 6/3 into c= tick, tick 2 B= V (a^2-c^2) =1, the equation is x^2/3+y^2/1=1 Two requirements area Obviously, AB is the base of the triangle Simultaneous x^2/3+y^2/1=1 Y=x+1 x1=0 y1=1, x2=-1.5 solution, y2=-0.5., using the chord length formula V (1+k^2)) [x2-x1] (in brackets indicate absolute value) chord =3 root 2/2, the largest area for P It should be the largest distance to the string Assume that the distance from the p to the chord has been found maximum P parallel lines that make strings It can be found that the parallel line is an ellipse and the tangent is the largest The tangent chord is parallel, so the slope of the slope chord = Let y=x+m use the discriminant equal to 0 Obtain m=2, -2., combining graphs, m=-2.x=1.5, y=-0.5, P (1.5, -0.5), Three linear equation x-y+1=0, the use of point to line distance formula for the root 2/2, 1/2* area 2/2*3 2/2=3/4 tick tick, History The ellipse has some optical properties: the mirror of the ellipse (the axis of the ellipse is the axis) A solid figure that forms 180 degrees of rotation of an ellipse All its surfaces are made of reflective surfaces Hollow (hollow) can reflect the light emitted by a focal point to another focal point; the ellipse lens (some cross section is ellipse) has the effect of converging rays (also called convex lens) Presbyopic glasses, magnifier and hyperopia glasses are the lenses (these optical properties can be proved by reductio) Some history about conic sections: conic section discovery and research starting in ancient Greece Euclid, Archimedes, Apollonius, Pappus and other Research Masters are keen on the geometry of the conic sections And there are monographs to discuss its geometric properties Among them, the book "cone curve theory", which is published by Apollonius, is composed of eight parts Can be said to be the ancient Greek geometry of a fine for breaking reach the peak of perfection The study of this simple and perfect curve at that time Purely from a geometric point of view Discuss the curves that are closely related to the circle; their geometry is the natural extension of the geometry of the circle In those days, it was a purely philosophical exploration Neither hope nor expectation, they really play an important part in the basic structure of nature The matter lasted until the turn of sixteen and seventeenth Century Kepler the discovery of the three law of planetary motion shows the orbits of planets orbiting the sun It is an ellipse with the sun as its focus Kepler's law is a major breakthrough in the dreamtime of modern science It not only ushered in a new era of astronomy It is also the root of Newton's law of gravity Thus it can be seen Conic section is not only simplifying things like geometry One of the essentials of the natural selection of their basic rule is in nature Elliptical hand drawing (1) painting long axis AB Short axis CD AB and CD share each other vertically at O points (2) connecting AC (3) take O as the center of the circle OA takes the radius as the arc and the OC extension line at the E point (4) take C as the center of the circle CE takes the radius as the arc and the AC at the F point (5): AF vertical bisector CD extension line in G Cross AB at H (6) interception of H G the symmetric point for the O point H' G'(7): H H'is the center of the long axis The radius is HB and H'A respectively; G G'is the short axis primitive heart Take GC and G'D as radii respectively With a line or fine wire, pencil, 2 pins or pin ellipse method: first paint the length of the shaft cross line, with a dot in the long axis as the center to find 2 greater than the short axis radius, first pin or bolt head needle line fixed, another a bit line should not be fixed, with live line to find the length of the shaft with 4 vertices, this step requires multiple positioning, until just to fix the vertex anastomosis after the 2 point pen with live line, draw the ellipse:) using thin copper wire is best, because the line drawn greater flexibility not necessarily accurate! Simple properties of an ellipse The angle of the two long vertices of an ellipse and a short vertex is greater than the connecting line between the point at which the ellipse takes place and the two long vertices Hand drawn ellipse method two (mayue) FF', the focal length of the ellipse (Z) definition The long axis X (AB) and the short axis Y (CD) of the known ellipse consist of a long axis A at one end and a short axis Y as the radius The line of focus from the other section of the long axis B leads to the tangent of the arc To verify the formula for 2 V {(Z/2) ^2+ (Y/2) ^2}+Z=X+Z (plane and two point F, F'distance and equal to the constant 2A (2a>|FF'|) of the P point trajectory is called elliptic) Can the evolution of z= root x^2-y^2 (x>y>0) Z points F and F'are fixed points Take the line with a smaller coefficient of toughness and expansion Surround the line segment AF'or FB line, any length of group The length is fixed triangle perimeter The ellipse is formed by taking F and F'as the fixed point and taking the third point on the triangle as a fixed point and drawing arc Ellipse () function Function function: this function is used to draw an ellipse The center of the ellipse is the center of the circumscribed rectangle Draws an ellipse with the current brush An ellipse filled with the current brush Function prototypes: BOOL, Ellipse (HDC, HDC, int, nLeftRect, int, nTopRect, nRightRect, int, nBottomRect) Parameter: HDC: device environment handle NLeftRect: Specifies the X coordinate of the upper left corner of the bound rectangle NTopRect: Specifies the Y coordinate of the upper left corner of the bound rectangle NRightRect: Specifies the X coordinate of the lower right corner of the bound rectangle NBottomRect: Specifies the Y coordinate of the lower right corner of the bound rectangle Return value: if the function call succeeds The return value is nonzero; if the function call fails The return value is 0 Hyperbola Definition: the difference between the absolute value of the difference in the distance between the plane and the two points F1 and F2 is equal to the trajectory of a constant. This is called hyperbolic definition 1: In-plane The trajectory of the point at which the absolute distance between the two points is constant (less than the distance [1] between the two points) is called a hyperbola Definition 2: in-plane The ratio of the distance to a given point and a line is greater than 1 and is constant The locus of a number is called a hyperbola Definition 3: a plane, a circle, a cone When the cross section and the generatrix of the cone surface are not parallel And intersect the two cones of the cone The intersection line is called hyperbola Definition 4: in a rectangular coordinate system Two yuan two equation H (x, y) =ax^2+bxy+cy^2+dx+ey+f=0 satisfies the following conditions The image is hyperbolic 1., a, B, C are not all 0 2. b^2 - 4ac > 0 In analytic geometry in high school It is learned that the center of the hyperbola is at the origin Image about X Y axisymmetric case At this point the hyperbolic equation is reduced to: x^2/a^2 - y^2/b^2 = 1 The four definitions mentioned above are equivalent Important concepts and properties Some related concepts and properties of hyperbola are given from the point of view of pure geometry A hyperbola has two branches The two given point referred to in definition 1 is called the focus of the hyperbola The point of reference referred to in definition 2 is also the focus of the hyperbola The hyperbola has two foci In a given line defined 2 mentioned is called the hyperbolic. The ratio of the distance given to the given point to the given line in definition 2 The eccentricity is called the hyperbola The hyperbola has two foci Two. (Note: Although the definition 2 only mentioned a focal point and one line But given a focus on the same side A directrix and centrifugal rate can be defined according to the 2 and two hyperbola On both sides of the focus Hyperbolic alignment and same eccentricity are the same ) The intersection of a hyperbola with a two focal line A vertex called a hyperbola There are two hyperbolic asymptote Simple geometric properties of hyperbola The range of point 1, track: x > A, x = -a (focus on the X axis) or y = a, y = -a (focus on the Y axis) 2, symmetry: about axis and origin symmetry 3, vertex: A (-a, 0) A'(a, 0) At the same time AA'is called hyperbolic, AA' / =2a. and the real axis B (0, -b) B'(0, b) At the same time BB'is called hyperbolic imaginary axis and / BB' / =2b. 4, asymptote: The focus is on the X axis: y= + (b/a) X. Focus on the Y axis: y= + (a/b) X. conic P =ep/1-ecos when e>1. Hyperbolic representation Where p is the focus to the alignment distance Theta is the angle between the chord and the X axis Make 1-ecos theta =0 to find theta This is the asymptote of inclination Theta =arccos (1/e) Theta =0 The P =ep/1-e, P cos 0 =ep/1-e x= Theta =PI The P =ep/1+e, P cos 0 =-ep/1+e x= The two X are the abscissa of the hyperbola Find the abscissa of their midpoint (hyperbola, center, abscissa) X= [(ep/1-e) + (-ep/1+e)] /2 (note that simplify) Linear P cos = [(ep/1-e) + /2 (-ep/1+e)] It's a hyperbola, a symmetrical axis Attention is the symmetrical axis that does not intersect the curve The line is clockwise rotation of the PI/2-arccos (1/e) to obtain the asymptotic equation of angle Set the angle after rotation is theta Is'= theta theta (1/e) [PI/2-arccos] Then the theta theta = '+ [PI/2-arccos (1/e)] Bring in: P cos{0 + [PI/2-arccos (1/e)}= [] (ep/1-e) + /2 (-ep/1+e)] Namely: P sin [arccos (1/e) - 0 '] = [(ep/1-e) + /2 (-ep/1+e)] Now theta can be used instead of theta Get the equation: P sin [arccos (1/e) - 0] = [(ep/1-e) + /2 (-ep/1+e)] This is to certify that x^2/a^2-y^/b^2=1 at a point on the hyperbola asymptote in Let M (x, y) be the point of the hyperbola in the first quadrant be Y= (b/a) * (x^2-a^2) (x>a) Because the x^2-a^2< a root x^2= "bx/a" a) V (x^2-a^2) That is, y< a> therefore The points in the first quadrant of the hyperbola are below the line y=bx/a According to symmetry, the same is true of the second, third, fourth quadrant 5, centrifugal rate: The first definition: e=c/a and E (1 ~ +). Second definition: the hyperbolic point on the P to point to the line, PF, and P from F (the corresponding directrix) distance ratio D is equal to the rate of E. centrifugal hyperbola (D / PF / /d (P) line to line (the line) from =e) 6, hyperbolic radius formula (P point (x, y) to the focus distance) on the conic curve Left: r= / ex+a / focal radius Right focal radius: r= / ex-a / 7, equal axis hyperbola The real axis hyperbola and imaginary axis length: 2a=2b and e= * 2 Then the asymptote equation is: y= + X (both focus on the X axis or Y axis) 8 conjugate hyperbola The real axis of the hyperbola S'is the imaginary axis of the hyperbola S, and the imaginary axis of the hyperbola S' is the real axis of the hyperbola S The hyperbola S'and the hyperbola S are conjugate hyperbola Geometric expression: S: (x^2/a^2) - (y^2/b^2) =1, S': (y^2/b^2) - (x^2/a^2) =1 Features: (1) a total of asymptote (2) the focal length is equal (3) the reciprocal of the square of the eccentricity of the two hyperbola is equal to 1 9, alignment: focus on the X axis: x= + a^2/c The focus is on the Y axis: y= + a^2/c 10 、 path length: (conic curve (except circle) Crossing the focus and perpendicular to the axis of the string D=2b^2/a 11, the focus of the chord length formula: D=2pe/ (1-e^2cos^2 theta) 12, the chord length formula: D = V (1+k^2) |x1-x2| (1+k^2) = V (x1-x2) = ^2 (1+1/k^2 = |y1-y2|) tick tick (1+1/k^2) (y1-y2) ^2 is derived as follows: The slope formula of the line is: k = (Y1 - Y2) / (x1 - x2) Get Y1 - y2 = K (x1 - x2) or X1 - x2 = (Y1 - Y2) /k The distance between two points are calculated formula: |AB| = v [(x1 - x2) 2 + (Y1 - Y2)] 2 You'll have to tidy up a little: |AB| = |x1 - x2| V (1 + K2) or |AB| = |y1 - y2| V (1 + 1/k2) Hyperbolic standard formula and inverse scaling function X^2/a^2 - Y^2/b^2 = 1 (a>0, b>0) The standard inverse proportion function is xy = C (C = 0) But the inverse scaling function is really the hyperbolic function that is rotated Because the symmetry axis of xy = C is y=x, y=-x, and the symmetry axis of X^2/a^2 - Y^2/b^2 = 1 is the X axis Y axis So it should be rotated 45 degrees A rotation angle for a (a = 0, clockwise) (a is the slope angle of the hyperbolic progressive line) Is there X = xcosa + ysina Y = - xsina + ycosa Take a = Pi /4 be X^2 - Y^2 = (xcos (PI /4) + ysin (PI /4)) ^2 - (xsin (PI /4) - ycos (PI, /4)) ^2 (v = 2/2 + X 2/2 y ^2 (V) - 2/2 X - 2/2 y tick tick) ^2 (v = 4 2/2 (x) 2/2 V y) = 2xy. And xy=c therefore X^2/ (2C) - Y^2/ (2C) = 1 (c>0) Y^2/ (-2c) - X^2/ (-2c) = 1 (c<0) This proves that Inverse scale function is actually hyperbolic function. It is just another form of hyperbola in plane Cartesian coordinate system Hyperbola, focus triangle, area formula If the angle F1PF2=, S F1PF2=b^2; cot (/2 0) Example: known F1 and F2 are hyperbolic C:x^2; -y^; =1's left and right focus Point P on C Angle F1PF2=60 degrees Then the distance from P to X axis is more Less? Solution: S F1PF2=b^2 by the triangle area hyperbolic focus formula; cot (0 /2) * cot30 ~ =1 The distance from P to X axis is h S F1PF2=1/2 * F1F2 * h=1/22 * h= * 3 V 2 H= root 6/2 parabola Definition The locus (or set) of a point in a plane that is equal to a point F and F but equal to L, is called a parabola The fixed point F is not in the straight line, moreover, F is called the parabola focal point" L called "parabolic directrix" Line focus to the parabolic distance defined as "Jiao quasi distance", represented by P p>0. Insert a cutting plane into a cone in parallel with the ground You can get a circle If the plane is tilted until it is parallel to its side You can make a parabola Standard equation The standard equation of parabola has four: Right opening parabola: y^2=2px Left open parabola: y^2= -2px Upper opening parabola: x^2=2py Lower opening parabola: x^2= -2py P is the focal distance (p>0) In parabolic y^2=2px The focus is (p/2 0) Alignment of L equation is x= -p/2; y^2= -2px in parabola The focus is (-p/2 0) Alignment of L equation is x=p/2; in parabolic x^2=2py The focus is (0) P/2) Alignment of L equation is y= -p/2; x^2= -2py in parabola The focus is (0) -p/2) The L equation is y=p/2; Correlation parameter (parabolic for right) Eccentricity: e=1 Focus: (p/2 0) Alignment of l:x=-p/2 equation Apex: (0 0) Path: 2P; definition: conic curve (in addition to circle) A focus and chord perpendicular to the axis of the domain (X = 0) Range (Y, R) Analytic solution Take the focus on the X axis as an example Know P (x0 Y0) Call for y^2=2px There is y0^2=2px0 * 2p=y0^2/x0 L y^2= (y0^2/x0) x parabola optical properties The beam of a light reflected by a parabola parallel to the axis of a parabola Area and arc length formula Area Area=2ab/3 Arc length Arc, length, ABC V = (b^2+16a^2) /2+b^2/8a (LN (4a+ * (b^2+16a^2)) /b) Other Parabola: y = ax^2 + BX + C (a = 0) That is, y equals the square of ax plus BX plus C When a > 0, open up A < 0, speak down At C = 0, the parabola passes past the origin When B = 0, the parabolic symmetry axis is the Y axis There is also vertex y = a (X-H) ^2 + k That is, y is equal to a times the square +k of (X-H) H is the X of vertex coordinates K is the Y standard form parabola of vertex coordinates, in X0 The tangent line of the Y0 point is: yy0=p (x+x0) Generally used for maximum and minimum values Parabolic standard equation: y^2=2px It is said that the focus of the parabola in X positive half axis, the focus of coordinates (p/2,0) alignment equation is x=-p/2 Since the focus of the parabola can be on any half axis, there are standard equations y^2=2px, y^2=-2px, x^2=2py, x^2=-2py Symmetry problem solving We know Parabolic y = ax^2 + BX + C (a = 0) is the axis of symmetry Its symmetry axis is linear x = - b/ 2A Its vertices are on the axis of symmetry When solving parabolic problems If we can use the symmetry of parabola skillfully A simple solution is often given Example 1 the symmetric axis of a known parabola is x =1 The parabola and the Y axis are at the point (0 3) The distance between the intersection point of the X axis and the two axis is 4 An analytic expression for this parabola is given The analytic parabolic formula is y = ax^2 + BX + C If conventional solution We need to solve the three element equation set about a, B and C The deformation process is quite complicated; if the symmetry of the parabola is used skillfully The solution is simple Because the symmetry axis of the parabola is x =1 The distance between the intersection point of the X axis and the two axis is 4 The symmetry of a parabola can be seen It is attached to the X axis at A (-1) 0) and B (3) 0) two points Thus, the analytic formula for parabola can be y = a (x+1) (x-3) Also because the parabola and the Y axis are at the point (0 3) So 3 = -3a So a =-1 R = - Y (x+1) (x-3) That is Y = - x^2 + 2x +3 Example 2 the known parabola passes through A (-1 2) and B (3) 2) two points The ordinate of its vertex is 6 Asks the value of y when x =0 The analysis requires the value of y when x =0 As long as the parabolic equation can be solved The symmetry of a parabola can be seen A (-1 2) and B (3) 2) two points are the symmetrical points on the parabola From this we can see The symmetrical axis of the parabola is x = 1 So the apex of the parabola is (1) 6) Thus, the analytic formula for parabola can be y = a (x-1) 2+ 6 Because of the point (-1 2) on a parabola So 4A + 6 = 2 So a = -1 R = - Y (x-1) ^2+ 6 That is Y = - x^2 + 2x +5 L when x =0 Y = 5 Example 3 the known parabola is the intersection of the X axis two and the distance between A and B is 4 Cross the Y axis at point C Its vertex is (-1 4) For the delta area of ABC Analysis of the requirements of the area of delta ABC Just find the coordinates of the point C to this end An analytic formula requiring parabola The problem is known The symmetrical axis of the parabola is x = -1 The symmetry of a parabola can be seen The coordinates of A and B are -3 0), (1) 0) Therefore, the analytic formula of parabola can be y = a (x+1), ^2+, 4[or y = a (x+3) (x-1)] Dreams (1 points 0) on a parabola * 4A + 4 = 0 * a = -1 R = - Y (x+1) 2+ 4 That is Y = - X2 - 2x +3 Coordinate point for C (0 * 3) S ABC = 1/2 * * * (4 * 3) = 6 Example 4 is known as parabola y = AX2 + BX + C, and the ordinate of vertex A is 4 Cross the Y axis at point B To the X axis, at C, D, two And -1 and 3 are the two roots of the equation AX2 + BX + C =0 Find the area of the quadrilateral ABCD The analysis requires the area of quadrilateral ABCD Find the coordinates of A and B at two points to this end An analytic formula requiring parabola is required The problem is known The coordinates of C and D are -1 0), (3) 0) The symmetry of a parabola can be seen The symmetrical axis of the parabola is x = 1 So the coordinates of the vertex A are (1) 4) Thus the analytic formula for the parabola can be y = a (x-1), 2+, 4[or y = a (x+1) (x-3)] Some dreams (-1 0) on a parabola * 4A + 4 = 0 So a = -1 R = - Y (x-1) ^2+ 4 That is Y = - x^2 + 2x +3 Coordinate point for B (0 * 3) Link OA S Quad ABCD = S + S + AOB Delta BOC Delta S Delta AOD = 1/2 * 1 * 3+1/2 * 3 * 1+1/2 * 3 * 4=9 Relevant conclusions The parabolic y^2=2px (p>0) focus F is the straight line L with an inclination angle of theta, and the L intersects the parabola at A (x1, Y1) B (X2, Y2), yes X1*x2 = p^2/4, y1*y2 = -P^2, set up when the line is over focus The focus of the chord length: |AB| = x1+x2+P = 2P/[(sin 0) ^2] (1/|FA|) + (1/|FB|) = 2/P If OA is perpendicular to OB, then AB is over designated M (2P) 0) The focal radius: |FP|=x+p/2 (a parabolic point P to the focus of the F is equal to the distance to L distance.) The chord length formula: AB= V (1+k^2) * / x2-x1 / The delta =b^2-4ac The =b^2-4ac>0 has two real roots The =b^2-4ac=0 has the same two real roots The =b^2-4ac<0 no real roots By the vertical tangent to the parabola focus Is the distance from the focus to the point of tangency With the mean proportional to the vertex distance Defining solutions Example: F is known to be the focus of parabolic y^2=4x A (3 2) is a fixed point P is the fixed point on the parabola Find the minimum value of |PA|+|PF| and the coordinates of the P at this time Solution: a parabolic L. P PH group of L Pedal for H Another point A AH't L Pedal for H' Join parabola in P' Link P'F Then: |PA|+|PF|=|PA|+|PH| = |AH'|=|P'A|+|P'H|=|P'A|+|P'F| therefore The minimum value of |PA|+|PF| is |AH'| The alignment of x=-1 equation Therefore, the minimum value of |PA|+|PF| is 4 here The coordinates for P'are (1) 2)