§ 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 M ￠ corresponding to it . Make the following two conditions are satisfied:

    ( i ) O , M , M ￠ are collinear,

       ( ii ) OM × OM ￠ = r ² ,

This kind of point M ￠ is 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 M ￠ is symmetric, M is also the inversion point of M ￠ . Since r ² > 0 , both M and M ￠ are on the same side of O. The correspondence between M and M ￠ is 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:

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 ₂ ￠ must intersect at the inversion point M ￠ of M.   

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

直    径     PQ(通过椭圆中心的弦)

图 7.2

共轭直径     二直径斜率为，且满足

准    线     L₁和L₂(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

               

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

       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 , M ￠ make a straight line parallel to NN ￠ ; The area of ​​the quadrilateral is a constant 4 ab ( 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 )

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

                         

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) 

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 . 

7 ° Let MM ￠ , NN ￠ be the two conjugate diameters of the hyperbola , and draw straight lines parallel to NN ￠ through M , M ￠ respectively ; 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

 

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

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 .