Аналитическая геометрия. Шурыгин В.В. - 52 стр.

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A(a; 0) B(0; b)
` Ox
1
Ox
2
a b
` `
x
1
a
+
x
2
b
= 1.
` A
1
x
1
+ A
2
x
2
+ A
3
= 0
P (x
i
P
) Q(x
i
Q
) ` α : R
2
→ R
α(x
1
, x
2
) = A
1
x
1
+ A
2
x
2
+ A
3
P (x
i
P
) Q(x
i
Q
)
` sgn α(x
i
P
) = sgn α(x
i
Q
)
P Q
` K ` P Q
λ = (P QK) > 0 x
i
K
=
x
i
P
+λx
i
Q
1+λ
K ∈ `
A
1
x
1
P
+ λx
1
Q
1 + λ
+ A
2
x
2
P
+ λx
2
Q
1 + λ
+ A
3
= 0,
α(x
i
P
) + λα(x
i
Q
) = 0 −
α(x
i
P
)
α(x
i
Q
)
= λ > 0
sgn α(x
i
P
) = −sgn α(x
i
Q
)
P Q ` S
` sgn α(x
i
P
) = −sgn α(x
i
S
) =
sgn α(x
i
Q
)
M(x
1
0
; x
2
0
)
` A
1
x
1
+ A
2
x
2
+ A
3
= 0
A
1
(x
1
− x
1
0
) + A
2
(x
2
− x
2
0
) = 0.
` x
1
−
7x
2
+5 = 0 ABC A(3; 1) B(−2; 4)
C(1; 0)
   óTÍtXxXWx ³T´µuÌ t uzTxêyÍõZ ¼/70 0[b*/8, 8-)10 A(a; 0) 0 B(0; b)
.*3*/*)*,09 .39;-M ` / -/9;0 1--3L0,68 Ox 0 Ox ‚a 0 b · -83*[10\                1                   2
-8/*16*;* .39;-M ` ,6 -/9N 1--3L0,68ƒ\ 8- 236b,*,0* .39;-M ` ;-|,-
[6.0/68? b b0L*
                              x
                                 +
                                    x
                                       = 1.
                                                        1       2
                                                                     ‚Æãƒ
(36b,*,0* ‚Æム,6[b6*8/9 œ”ŽŒŽ’
                               a    b
                                       ¡£’Š¦ ” ŠžŽ¢‰Ð=
Z± YêÍWµXux TͶ³u™ušxXWx wtSõ zuñxy uzXu¶Wzx™ëXu ³T´µuÌ Z
H2/8? [6L6,6 .39;69 ` / 236b,*,0*; A x + A x + A = 0 0 .636 8-)*1
                                           {
                                                                            1               2

P (x ) 0 Q(x ) \ ,* 7*|6}0N ,6 .39;-M ` = b*L*; O2,1 0J α : R → R \
                                                                        1           2                   3
      i           i                                                                                                   2
.-76:69 α(x , x ) = A x + A x + A =
      P
              1
                  Q
                      2            1                2
   ˜Txw™ušxXWxZ ¨Š¤‰Œ P (x ) Œ Q(x ) ‹Ž»ž ¡Š Šªœ žŠŠœ Šž ¡£’Š¦
                               1                2           3


     ª                ª ª
                                            i               i

` žŠ©  Œ žŠ‹Ÿ‰Š žŠ© ¥ ‰Š©  sgn α(x ) = sgn α(x ) =
                                            P               Q
                                                                    i                           i
   éuyÍêÍzx™ë¶ztuZ H3*L.-7-|0; /,6)676\ )8- P 0 Q 7*|68 .- 36[,*   P                           Q


/8-3-, -8 .39;-M ` = H2/8? K · 8-)16 .*3*/*)*,09 .39;N ` 0 P Q = „-:L6
λ = (P QK) > 0 0 x =          = „61 161 K ∈ ` \ 8-
                           i           xiP +λxiQ
                           K              1+λ

                             x1P + λx1Q      x2P + λx2Q
                          A1            + A2            + A3 = 0,
                                1+λ             1+λ
)8- ~1b0b67*,8,- 8-;2\ )8- α(x ) + λα(x ) = 0 \ -812L6 − = λ > 0 \
                                                        i                   i                               α(xiP )

/7*L-b68*7?,-\ sgn α(x ) = −sgn α(x ) =                 P                   Q                               α(xiQ )

   H2/8? 8*.*3? P 0 Q 7*|68 .- -L,2 /8-3-,2 -8 ` \ 0 S · ,*1-8-369 8-)K
                                   i                        i
                                   P                        Q


16\ 7*|6}69 .- L32:2J /8-3-,2 -8 ` = „-:L6 sgn α(x ) = −sgn α(x ) =                                 i                     i

sgn α(x ) = 
                                                                                                    P                     S
          i
   ðÍwÍñÍ ±ÊZ ­-/86b08? 236b,*,0* .39;-M\.3-N-L9}*M )*3*[ 8-)12 M (x ; x )
          Q
                                                                                                                              1   2
.63677*7?,- L6,,-M .39;-M ` / 236b,*,0*; A x + A x + A = 0 =                            1               2
                                                                                                                              0   0


   àxÿxXWxZ ˆ)*b0L,-\ )8- 0/1-;-* 236b,*,0* 0;**8 b0L                           1                   2        3




                               A1 (x1 − x10 ) + A2 (x2 − x20 ) = 0.
   ðÍwÍñÍ ±Z ˆ.3*L*708? .-7-|*,0* .39;-M ` \ 0;*J}*M 236b,*,0* x −                                                           1

7x +5 = 0 \-8,-/08*7?,- 83*2:-7?,016 ABC / b*3I0,6;0 A(3; 1) \ B(−2; 4) \
  2

C(1; 0) =
                                  ¾ï

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