§ 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 subconnected 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:
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 _{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F_{1}H)
焦点半径 r_{1}, r_{2}(椭圆上一点(x, y)到焦点的距离)
r_{1 }= a  ex， r_{2 }= 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 L_{1}和L_{2}(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 semiconjugate 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).
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 semiaxis 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 overfocus 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 semiconjugate diameter .
Figure 7.10 
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
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 Secondrate 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 Secondrate 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 )
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 .