§ 5 Quadratic Curve

 

1.      Circle

 

[ Circle equation, center and radius ]

Equations and Graphics       

Center and Radius

 

  x 2 + y 2 = R

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    

焦点参数     (等于过焦点且垂直于长轴的弦长之半,即F1H)

焦点半径     r1, r2(椭圆上一点(x, y)到焦点的距离)

r1 = a - ex r2 = a + ex

         PQ(通过椭圆中心的弦)

7.2

共轭直径     二直径斜率为,且满足

    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

       

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.[1051]  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 .

 


 [1051]

Original text