§ 5 Quadratic Curve

1.      Circle

[ Circle equation, center and radius ]

 Equations and Graphics Center and Radius x 2 + y 2 = R 2  or    ( parametric equation, t is the angle between the moving diameter OM and the positive direction of the x -axis ) Center G (0,0)             radius r = R ( x - a ) 2 +( y - b ) 2 = R 2   or    ( parametric equation, t is the angle between the moving diameter OM and the positive direction of the x -axis ) Center G ( a , b )            radius r = R x 2 + y 2 + 2 mx + 2 ny + q = 0       m 2 + n 2 > q r 2 + 2 r ( m cos t + n sin t ) + q = 0 ( polar equation ) Center G ( - m , - n )      radius r 2 - 2 rr 0 cos( j - j 0 ) + r 0 2 = R 2           ( polar equation ) Center G ( r 0 , j 0 )            radius r = R x 2 + y 2 = 2 Rx    or r = 2 R cos j       ( Polar Coordinate Equation ) Center G ( R , 0)             radius r = R x 2 + y 2 = 2 Ry  or r = 2 R sin j        ( Polar Coordinate Equation ) Center      G(0, R )    radius r = R

[ Circle tangent ]

The equation of the tangent to a point M ( x 0 , y 0 ) on the circle x 2 + y 2 = R 2 is

x 0 x + y 0 y = R 2

The equation of the tangent to a point M ( x 0 , y 0 ) on the circle x 2 + y 2 + 2 mx + 2 ny + q = 0      is

x 0 x + y 0 y + m ( x + x 0 ) + n ( y + y 0 ) + q = 0

[ Intersection of two circles, circle bundle and root axis ]

 Equations and Graphics Formula and Explanation The intersection of two circles C 1 x 2 + y 2 + 2 m 1 x + 2 n 1 y + q 1 = 0   C 2 x 2 + y 2 + 2 m 2 x + 2 n 2 y + q 2 = 0          The intersection angle of two circles is the angle between their two tangents at the intersection In the formula, q represents the intersection angle of the two circles C1 and C2 , because the coordinates of the intersection point are not included in the formula, so the two intersection angles at the two intersection points must be equal .        The two circles C1 and C2 are orthogonal to the condition that               2 m 1 m 2 + 2 n 1 n 2 - q 1 - q 2 = 0 Circle bundle × root axis of two circles C l C 1 + l C 2 = 0 ( l is a parameter )   or   ( l + 1)( x 2 + y 2 ) + 2( m 1 + l m 2 ) x + 2( n 1 + l n 2 ) y + ( q 1 + l q 2 ) = 0 The root axis equation is 2( m 1 - m 2 ) x + 2( n 1 - n 2 ) y + ( q 1 - q 2 ) = 0 For a certain value of l ( l 1 - 1) , C l represents a circle . When l takes all values ​​( l 1 - 1) , the whole of the circles represented by C l is called a circle bundle . l = - 1 When , it is a straight line, which is called the root axis of the two circles C1 and C2. The root axis is perpendicular to the connecting center line of C1 and C2 , and the center of any circle C1 in the bundle is at the center of C1 and C2 . connected to the central line, and the ratio of the sub-connected central lines is equal to l . ( a ) If C 1 and C 2 intersect at two points M 1 , M 2 , then all circles in the bundle pass through the two intersection points M 1 , M 2 , and their root axis is their common chord. At this time circular bundle is called Coaxial circle system ( Fig. ( a )). ( b ) If C1 and C2 are tangent to a point M , then all circles in the bundle are tangent , and the root axis is the common tangent at the point M ( Figure ( b ) ) . ( c ) If C1 and C2 do not intersect, all circles in the bundle do not intersect, and the root axis does not intersect with all circles in the bundle ( Figure ( c ) ) .   Draw tangents to two circles C1 and C2 from point P , and the locus of point P with equal tangent lengths is the root axis . The root axis of the two concentric circles is a straight line from the common center to infinity . Among the three circles The root axes of each pair of circles ( three in total ) intersect at a point, which is called the root center . If the three circle centers are collinear, the root center is at infinity .

[ Inversion ]    Let C be a certain circle, O be the center of the circle, r be the radius ( Fig. 7.1) , for any point M on the plane , there is a point Mcorresponding to it . Make the following two conditions are satisfied:

( i ) O , M , Mare collinear,

( ii ) OM × OM= r 2 ,

This kind of point Mis called the inversion point of point M about the fixed circle C , C is called the inversion circle, O is the inversion center, and r is the inversion radius .

Since the relationship between M and Mis symmetric, M is also the inversion point of M. Since r 2 > 0 , both M and Mare on the same side of O. The correspondence between M and Mis called about Inversion of definite circle C.

Taking O as the origin, the corresponding equations of all inversion points M ( x , y ) and M( x, y) are

Inversion has the properties:

 Figure 7.1

1 ° A straight line not passing through the inversion center becomes a circle passing through the inversion center .

2 ° A circle passing through the inversion center becomes a straight line that does not pass through the inversion center .

3 ° becomes itself by a straight line through the center of the inversion .

4 ° A circle that does not pass through the inversion center becomes a circle that does not pass through the inversion center .

The 5 ° inversion circle becomes itself .

6 ° The circle orthogonal to the inversion circle becomes itself, and its inverse is true .

7 ° If the two curves C 1 , C 2 intersect at a point M , the inversion curves C 1, C 2must intersect at the inversion point Mof M.

8 ° If the two curves C 1 , C 2 are tangent at a point M , then the inverted curves C 1, C 2must be tangent at the inversion point M ￠ of M.

The intersection angle of the two curves of 9 ° is unchanged under the inversion . It can be seen that the inversion is a conformal transformation .

2.      Ellipse

1. Basic Elements of Ellipse

Main axis ( symmetry axis )

Vertices A , B , C , D

Ellipse center G

Focus F 1 , F 2

focal length

Eccentricity _

compression factor

r1 = a - ex r2 = a + ex

PQ(通过椭圆中心的弦)

7.2

线     L1L2(平行于短轴，到短轴的距离为)

2．椭圆的方程、顶点、中心与焦点

 方  程  与  图  形 vertex · center · focus ( standard equation ) or ( parametric equation, t is the angle between the radius of the concentric circle ( radius a , b ) corresponding to point M and the positive direction of the x -axis ) Vertices A , B ( ± a , 0)                    C , D (0, ± b ) Center G (0,0)      Focus F 1 , F 2 ( ± c ,0) or       ( t same as above ) Vertices A , B ( g ± a , h )             C , D ( g , h ± b ) Center G ( g , h )      Focus F 1 , F 2 ( g ± c , h ) Vertices A , B (0, ± a )             C , D ( ± b , 0) Center G (0, 0)      Focus F 1 , F 2 (0, ± c ) , e < 1  ( Polar coordinate equation, the pole is located at the focus of the ellipse, the polar axis is the ray from the focus to the nearest vertex, j is the polar angle, p , e are as described above ) long axis      short axis      focal length

3. properties of an ellipse

A 1 ° ellipse is the locus   ( r 1 + r 2 = 2 a ) of a moving point M whose distances to two fixed points ( i.e. the focal points ) have a constant sum ( i.e. the major axis ) .

A 2 ° ellipse is also the locus of a moving point M whose ratio of the distance to a certain point ( ie, one of the focal points ) to a certain straight line ( ie, a directrix L ) is a constant ( ie, eccentricity ) less than 1 ( MF 1 / ME 1 = MF 2 / ME 2 = e ).

A 3 ° ellipse is obtained by compressing a circle with radius a along the y -axis in proportion ( ie, the compressibility factor ) .

The equation of the tangent ( MT ) at a point M ( x 0 , y 0 ) on a 4 ° ellipse is

The tangent line bisects the outer angle (i.e. ∠ F 1 MH ) between the two focal radii of point M ( i.e. a = b , ) , and the normal MN of point M bisects the inner angle ( i.e. F 1 MF 2 ) ( Figure 7.3 ) .

If the slope of the tangent ( MT ) of the ellipse is k , then its equation is

 Figure 7.3

The positive and negative signs in the formula represent the two tangents at the two ends of the diameter .

 Figure 7.4

Any diameter of a 5 ° ellipse bisects the chord parallel to its conjugate diameter ( Figure 7.4)

If the lengths of the two conjugate diameters are 2 a 1 and 2 b 1 respectively , and the included angles ( acute angles ) between the two diameters and the long axis are a and b respectively , then a 1 b 1 sin( a + b ) = ab

a 1 2 + b 1 2 = a 2 + b 2

The product of the focal radii of any point M on a        6 ° ellipse is equal to the square of its corresponding semi-conjugate diameter .

7 ° Let MM, NN ￠ be the two conjugate diameters of the ellipse , through M , Mmake a straight line parallel to NN; The area of ​​the quadrilateral is a constant 4 ab ( Figure 7.5).

 Figure 7.5

4. Calculation formula of each quantity of ellipse

 Ellipse quantities Calculation formula [ radius of curvature ]     R where r 1 , r 2 are the focus radius , p is the focus parameter , a is the angle between the focus radius of the point M ( x , y ) and the tangent . In particular , the curvature radius of the vertex               , [ arc length ] = where e is the eccentricity [ perimeter ]     L In the formula ,     set , then            or [ area ]     S sector ( OAM ) area        arcuate ( MAN ) area        Ellipse area S = p ab [ geometric center of gravity ]        G Oval G and O coincide _      half oval         ( a , b are the semi-axis lengths of the ellipse ) [ Moment of inertia ]       J The axis of rotation of the ellipse passes through the b axis        where m is the mass

3.      Hyperbola

1. Basic Elements of Hyperbola

Main axis ( symmetry axis )

 Figure 7.6

Vertices A , B

Center G

Focus F 1 , F 2

Focal length F 1 F 2 = 2 c ,

Eccentricity _

Focus parameter ( equal to the sum of the chord lengths that are over-focus and perpendicular to the real axis

half , i.e. F 1 H )

Focus radius r 1 , r 2        ( the distance from a point ( x , y ) on the hyperbola to the focus ,

i.e. MF 1 , MF 2 )

r 1 = ± ( ex - a ), r 2 = ± ( ex + a )

Diameter PQ ( chord through center )

Conjugate diameter two diameter slopes are k , k, and satisfy

Directives L 1 and L 2          ( perpendicular to the real axis , the distance from the center )

2 . Equation, vertex, center, focus, and asymptotes of a hyperbola

 Equations and Graphics vertex, center, focus, asymptote ( standard equation ) or        ( parametric equation ) or Vertices A , B ( ± a ,0)        Center G (0,0)        Focus F 1 , F 2 ( ± c ,0)               asymptote _ ( and form a conjugate hyperbola ) vertex        center        focus             asymptote _ vertex          center   focus                 asymptote _ equation with graphics vertex, center, focus, asymptote ( Polar coordinate equation . The pole is located at a focal point, and the polar axis is the ray from the focal point back to the vertex, p , e are as described above . From this equation, only one can be determined, and the other can be obtained by symmetry ) real axis     imaginary axis            focal length ( equiaxed hyperbola ) vertex           center         focus                   ( same sign when k > 0 , different sign when k < 0 ) shaft length         asymptote _ ( equiaxed hyperbola ) Vertices          ( same sign when D < 0 , different sign when D > 0 )    center         shaft length          asymptote _

3. Properties of Hyperbola

A 1 ° hyperbola is the locus of a moving point M whose distance to two fixed points ( focal points ) is a constant difference ( equal to the real axis 2 a ) ( so that each point belongs to one branch of the hyperbola, and each point belongs to the other. one ).

A 2 ° hyperbola is also the locus ( ) of the moving point M where the ratio of the distance to a certain point ( one of the focal points ) to the distance to a certain straight line ( directive line L 1 ) is a constant ( ie eccentricity ) greater than 1 .

The equation of the tangent ( MT ) at a point M on a 3 ° hyperbola is

 Figure   7.8

It bisects the interior angle ( ie ) between the radii of the two focal points at point M , while the normal MN at point M bisects the outer angle ( ie ) ( Figure 7.7) .

If the slope of the tangent of a hyperbola is k , then the equation of its tangent is

The positive and negative signs in the formula represent the two tangents at the two ends of the diameter .

4 ° The tangent line segment TT1 between the two asymptotes is bisected by the tangent point M ( TM = MT1 ) , and

D OTT 1 area ,

Area of ​​parallelogram OJMI ( shaded area in Figure 7.8 )

Any diameter of a 5 ° hyperbola bisects the chord parallel to the conjugate diameter ( Figure 7.9)

 Figure   7.9

If the lengths of the two conjugate diameters are 2 a 1 , 2 b 1 respectively , and the included angles ( acute angles ) between the two diameters and the real axis are a and b respectively ( a < b ) , then

The product of the focal radii of any point M on the 6 ° hyperbola is equal to the square of its corresponding semi-conjugate diameter .

 Figure   7.10

7 ° Let MM, NNbe the two conjugate diameters of the hyperbola , and draw straight lines parallel to NN through M , Mrespectively ; The area of ​​a parallelogram is a constant 4 ab ( Figure 7.10).

4. The formula for calculating the quantities of the hyperbola

 hyperbolic quantities Calculation formula [ radius of curvature ]      R where r 1 , r 2 are the focal radius, p is the focal parameter, a is the angle between the focal radius of the point M ( x , y ) and the tangent, in particular, the curvature radius of the vertices A , B hyperbolic quantities Calculation formula [ arc length ] = where e is the eccentricity [ area ]      S The area of ​​the bow ( AMN ) :               Area of ​​OAMI : Here OI , OJ are asymptotes, MI // OJ

4.      Parabola

 Figure 7.11

1. basic elements of a parabola

Axis AB of the parabola

vertex A

Focus F

The focus parameter p ( equal to overfocus and perpendicular to the axis

half the length of the string CD )

Focus radius MF ( a point on the parabola to the focus

distance )

Diameter EMH ( direction parallel to the axis of the parabola)

line )

Directrix L ( perpendicular to the axis of the parabola, the distance from the vertex A equals , and the distance from the focus F equals p )

2. Equations, vertices, focus and directrix of a parabola

 Equations and Graphics Vertex · Focus · Directive ( standard equation )       or                      ( Polar coordinate equation, the pole is located at the focus F , the polar axis coincides with the axis of the parabola, and faces away from the vertex ) Vertex A (0, 0)         focus         alignment _ Vertex A (0, 0)         focus         alignment _ Equations and Graphics Vertex · Focus · Directive Vertex A (0, 0)         focus             alignment _ Vertex A (0, 0)         focus             alignment _ Vertex A ( g , h )         focus         alignment _ Vertex A ( g , h )         focus        alignment _ vertex         ( When a > 0 , the opening is up,   When a < 0 , the opening is down ) focus parameter         Intersection with the x -axis               vertex             focus parameter

3. properties of a parabola

 Figure   7.12

A 1 ° parabola is the locus of a moving point M ( MF= ME ) whose distance to a certain point F ( the focal point ) is equal to the distance to a certain straight line L ( the directrix ) ( Figure 7.12)

The equation of the tangent MT at a point on a 2 ° parabola is

It bisects the angle ( D FMG ) between the focal radius of point M and the diameter ( D FMT = D TMG ) , and all chords parallel to the tangent MT are bisected by the diameter of point M ( PI = IQ ).

If the slope of the tangent to a parabola is k , then the equation of its tangent is

The angle between any two tangents of a 3 ° parabola is equal to half the angle between the focal radii of the two tangent points .

4 ° From the focus F , draw the perpendicular to the tangent of the parabola at point M , then the trajectory of the foot is the tangent at the vertex .

4. The formula for calculating the quantities of the parabola

Parabolic quantities

Calculation formula

[ radius of curvature ]

## R

where a is the angle between the tangent of the point M ( x , y ) and the main axis, and n is the length of the normal MN . In particular, the radius of curvature of the vertex R 0 = p

[ arc length ]

=

[ area ]

S

The area of ​​the arc ( MOD ) = the area of ​​the parallelogram ( MBCD )

which is

Here MD is the bow chord length , CD is parallel to the major axis , BC is tangent to the parabola ,

h is the height of the parallelogram ( that is, the arch height ), in particular ,

[ geometric center of gravity ]

## G

The center of gravity of the bow ( MOD )

( BC is parallel to MD , P is the tangent point , PQ is parallel to Ox )

5.      General quadratic curve

1 . General Properties of Quadratic Curves

The ellipse, hyperbola, parabola, etc. listed above , their equations are quadratic about x , y , and the general quadratic equation about x, y is in the form of

The curve it represents is called a general quadratic curve . Here are some common properties of them .

[ Intersection point of a straight line and a quadratic curve A straight line and a quadratic curve intersect at two points ( real , imaginary , coincident ).

[ Diameter and center of quadratic curve The midpoint of the chord of a quadratic curve parallel to the known direction is on a straight line , and it is called the diameter of the quadratic curve , which bisects a certain set of chords . The number of directions is a , b , the equation of the diameter is

or rewritten as

It can be seen that the diameters of the quadratic curve form a bundle of straight lines . Any diameter in the bundle passes through the intersection of the following two straight lines :

1 ° ie .

At this time, all the diameters of the quadratic curve pass through the same point , which is called the center . This kind of curve is called a centered quadratic curve . The coordinates of the center are

2 ° ie

(i)    At this time, the curve has no center ;

(ii)  At this time, the curve has an infinite number of centers , that is, the centers are on the same line ( center line ).

These two curves are called centerless quadratic curves .

[ Main axis ( or axis of symmetry ) of quadratic curve]  If the diameter is perpendicular to the chord bisected by it , it is called the main axis ( axis of symmetry ) of the quadratic curve. The concentric quadratic curve has a real main axis ; the centered quadratic curve A curve has two real major axes , they are perpendicular to each other , and the intersection is the center .

[ Tangent and normal of quadratic curve

The equation of the tangent to a point on a quadratic curve is

The line perpendicular to the tangent of the quadratic curve at point M is called the normal at point M , and its equation is

2 . Invariants of Quadratic Curves

From the equation of the general quadratic curve

(1)

The following three functions are composed of the coefficients of :

It is called the invariant of the quadratic curve , that is, after the coordinate transformation , these quantities are unchanged . The determinant D is called the discriminant of the quadratic equation (1) .

3 . Standard Equations and Shapes of Quadratic Curves

 invariant _ Standard equation after coordinate transformation Curve shape have Heart two Second-rate song String in the formula      A , C are characteristic equations     two characteristic roots of ellipse virtual ellipse A pair of imaginary lines with a common real point hyperbola intersect two lines without Heart two Second-rate song String in the formula parabola two parallel lines coincident two straight lines a pair of imaginary straight lines

4 . Several cases of quadratic curve

 A graphics Vertex·Center·Focus Parameters parabola vertex      focus parameter oval vertex        in center hyperbola

5. Conical section

Quadratic curves are all stubs that cut a regular conic surface with a plane . Therefore, quadratic curves are also called conic stubs (Figure 7.13 )

When cutting a regular cone with a plane P , if P does not pass through the top of the cone and is not parallel to any generatrix , the sectional line is an ellipse ; if P does not pass through the apex of the cone but is parallel to a generatrix , the sectional line is a parabola ; if When P does not pass through the top of the cone and is parallel to the two generatrixes , the sectional line is a hyperbola ; if P is perpendicular to the cone axis , the sectional line is a circle .

If P passes through the top of the cone , the ellipse becomes a point , the hyperbola becomes a pair of intersecting straight lines , and the parabola becomes a straight line tangent to P and the cone .