 . 
 


      ,           .





 . 





1. .  .      


    ,      ,        .       ,         .

          ,      ,     ,  Y   .       .






.1



 

     :     .  : (, ).       I, II, III, IV.     I   (),   ,    ,    II ,   ,   ,    III ,  ,   ,    IV ,  ,   .  ,    ,   ,  ,      ,     .

    ,         .         .

         ,   .      ? ( )   ?.        ( ?  ?).  ?, ?     .

         : x = r cos?, y = r sin? :













    


(


, 


)  


(


, 


).   :








    (  ):  +  +  = 0 (      ).

    ,   :  =  + b (  =   / , b =   / ).            , b              .  ,   :  = b,   :  = 0;  ,   :  = ,   :  = 0.




2.      .  .     .  .     


1.    


 (


, 


), 


 (


, 


), 


 (


, 


),       :








 (


  


) (


  


)  (


  x


) (


  


) = 0.

2.    


 (


, 


), 


 (


, 


),   ,     :








(


  


)(  


)  (  


)(


  


) = 0  (  


) / (


  


) = (  


) / (


  


).

3.    (


, 


)    L,    =  + .  ,    L  :

  


 = (  


).

  L    +  +  = 0,    ,    ,   (  


) + (  


) = 0.

 ,    L  :

  


 = (  


) / 



(  


) = 


  .

  L    +  +  = 0,    ,    (


, 


),    (  


) (  


) = 0.

4.    


 (


, 


), 


 (


, 


)  ,    +  +  = 0.      :

1) 


, 


       ,   (


 + 


 + )  (


 + 


 + )   ;

2) 


, 


       ,   (


 + 


 + )  (


 + 


 + )   ;

3)    


, 


    ,       (


 + + 


 + )  (


 + 


 + )   .

5.     ,      (


, 


),   .          


 =  (  


) (     ).

     : l(y  y


) = m(x  x


),  l, m       .

    L


  L


    (


 + 


 + 


) = 0  (


 + 


 + 


) = 0,   : m


(


 + 


 + 


) + m


(


 + 


 + 


) = 0.   L


  L


 ,   ,   ,    .

6.    (


, 


)  ,    +  +  = 0. d    :









3.  .   .  


   L    ( ,       )   ? (      ,        L).  ,    +  +  = 0:  








  ?








  C > 0  ,  C < 0   ,   = 0   ,    ,   .

   (   ) (c. .2): x cos? + y sin?  p = 0.       +  +  = 0.             


(      ).






.2



      :








   L,    .           /  +  / b = 1. :      /  +  / b = 1,       , b.

      ,    ,       .

      :     ,        . ,       , ,        *(


, 


),             *     


,   


 ..    = * + 


,  = * + 


  * =   


, * =   


(*   ).

      ?    ( ,     ; *, *      ):

x = x* cos?  y* sin?;

y = x* sin? + y* cos?



x* = x cos? + y sin?;

y* =  x sin? + y cos?.




4.  . . . . 


 L,      n   n.

   R       : 


 + 


 = R


,        (, b),  :

(  )


 + (  b)


 = R


.

  


 +  + 


 +  + D = 0  , ,        ,    


  


  ,  


 + 


  4D > 0 (        ).

  ,   


 +  + 


 +  + D = 0   : a = B / 2A, b = C / 2A, R


= (


 + 


  4D) / 4A


.

    (.3).






.3



 


   ,  


 = 2    ,  


 = 2b     (a > b)     ,  , 


, , 


   .  k = b / a   ? = 1  k = (a  b) / a   .            .

  : x


 / a


 + y


 / b


 = 1.

  :      (),        F, F


       2 (F


M + FM = 2a) (.4).






.4



 F  F


   ,   FF


   ,  FF


 = 2,   < .   ?         ? =  / .    , : k


 = 1  ?


.

     ,        F, F


        (.5). |F


M  FM| = 2a.  F, F


   ,  FF


 = 2c   . : c > a.

  : 


 / 


 + 


 / (


 


) = 1.      = bx / a  y =  bx / a (b


 = c


  a


).

          F ( )    PQ( ).      FC      .     .   : 


 = 2.






.5




5.   . 


       f(x, y, z) = 0.

  : +  + z + D = 0.  , , , D   ,    .

 D = 0   +  + z = 0  ,    .

  = 0,    ,   .     , ,    ,    ,     :   = 0   = 0   ,   = 0   = 0  z,   = 0   = 0  yOz.  Cz = 0   xOy, By = 0   xOz, Ax = 0   yOz.    :  /  +  / b + z / c = 1.     (


, 


, z


)  :








    


 + 


 + 


z + D


 = 0  


 + 


 + 


z + D


 = 0.  ?   :








   : 


/ 


 = 


/ 


 = 


 / 


 = D


 / D


.   : 


 / 


 = 


 / 


 = 


 / 


.   : 





 + 





 + 





 = 0.  ,      (


, 


, z


)  ,    +  + z + D = 0: (  x


) + (  y


) + (z  z


) + D = 0.  ,     


 (


, 


, z


), 


 (


, 


, z


), 


 (


, 


, z


):








 ,     


(


, 


, z


)  


(


, 


, z


)   ,   A


 + B


 + C


 + D = 0:








 ,    


 (


, 


, z


)     


 + 


 + 


z + D


 = 0  


 + 


 + 


z + D


 = 0,  :








  ,   :









6.  


       








 ()  : (x  x


) / m = (y  y


) / p = (z  z


) / q,     M


 (x


, y


, z


).  ?   ,   :








   : m


/ m


 = p


 / p


 = q


/ q


.    : m


m


 + p


p


 + q


q


 = 0.

   (x  x


) / m = (y  y


) / p = (z  z


) / q    +  + z + D = 0.     : Am + Bp + Cq = 0.     : A / m = B / p = C / q.    :








    x




  .


   .

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