Задачи по аналитической геометрии. Часть II. Игудесман К.Б. - 59 стр.

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r r < 0
(x−5)
2
9
+
(y−1)
2
4
− (z − 2)
2
= 0
x−6
3
=
y−2
0
=
z−8
4
x−6
9
=
y−2
8
=
z−8
20
x − 3z = 0
3x −2y −3z −18 = 0
x − 2y − 3z − 6 = 0
x−5
1
=
y−4
2
=
z−21
6
x−5
1
=
y−4
−2
=
z−21
14
10(x −5)
2
+ 20(x −5)(y −1) −34(y −1)
2
−55z
2
= 0
x−2
2
=
y−1
−1
=
z
1
x−4
2
=
y+2
1
=
z
2
A
2
a
2
+B
2
b
2
±C
2
c
2
= ±D
2
A
2
p ± B
2
q = 2CD x
2
+ y
2
= 13z
2
− 14z + 10
C(1, 1, −1); X
2
+ Y
2
+ Z
2
+ 2XY − 2Y Z + 6XZ − 1 = 0
2x + y + 6 = 0 7x − 35y + 22 = 0, 7x + 14y + 20 = 0
x−4y−2 = 0, x+4y−3 = 0 x+y−1 = 0, 3x+3y+13 = 0
z = 1; 2x − 3y = 0 4x + 2y − 5 = 0
x = 1, y = t, z = −t, 4XY + 4XZ − 1 = 0
X+Y +Z = 0 6x−2y+19 = 0, 2x+2y−1 = 0 7x+1 =
0 3x + 1 = 0, 3z − 2 = 0 7x + 17y + 19z + 19 = 0
X
2
9
+
Y
2
4
= 1 C(2, 3)
−
1
2
X
2
4
−
Y
2
9
= 1 C(1, 1)
2
3
x −y −1 = 0 x −4y + 2 = 0
X
2
= 2Y
X
2
16
+
Y
2
4
= 1
X
2
9
−
Y
2
25
= 1
(r  êîýôôèöèåíò ãîìîòåòèèè; åñëè r < 0, òî äîáàâëÿåòñÿ åùå ñèì-
ìåòðèÿ îòíîñèòåëüíî íà÷àëà êîîðäèíàò).                             130. 1) Ýëëèïñ; 2) ïàðà-
áîëà. 131. Ýëëèïòè÷åñêèé ïàðàáîëîèä èëè ãèïåðáîëè÷åñêèé ïàðàáî-
ëîèä. 132. Òàêèõ ïðÿìûõ ìîæíî ïðîâåñòè áåñ÷èñëåííîå ìíîæåñòâî;
                                                            (x−5)2   (y−1)2
èõ ãåîìåòðè÷åñêîå ìåñòî åñòü êîíóñ                            9    +    4   − (z − 2)2       = 0.
133.      x−6
           3    =   y−2
                     0      =   z−8
                                 4    è   x−6
                                           9        =   y−2     z−8
                                                         8 = 20 .      135. 1) x − 3z        = 0 è
3x − 2y − 3z − 18 = 0; ïðÿìàÿ ïåðåñåêàåò ïîâåðõíîñòü â äâóõ äåéñòâè-
òåëüíûõ òî÷êàõ; 2) äåéñòâèòåëüíûõ êàñàòåëüíûõ ïëîñêîñòåé ïðîâåñòè
íåëüçÿ; ïðÿìàÿ íå èìååò äåéñòâèòåëüíûõ òî÷åê ïðåñå÷åíèÿ ñ ïîâåðõ-
íîñòüþ; 3) x − 2y − 3z − 6 = 0; ïðÿìàÿ êàñàåòñÿ ïîâåðõíîñòè è ÷åðåç
íåå ìîæíî ïðîâåñòè òîëüêî îäíó êàñàòåëüíóþ. 136.                             x−5
                                                                              1    =   y−4
                                                                                        2    =   z−21
                                                                                                  6
è   x−5
     1    =   y−4
               −2   =   z−21
                         14 .       137. 1) Ýëëèïñîèä; 2) îäíîïîëîñòíûé ãèïåðáîëî-
èä. 138. Êîíóñ: 10(x − 5)2 + 20(x − 5)(y − 1) − 34(y − 1)2 − 55z 2 = 0.
139. x−2
      2 =
                 y−1
                  −1    =   z
                            1   è   x−4
                                     2    =   y+2
                                               1    = z2 . 141. 1) A2 a2 +B 2 b2 ±C 2 c2 = ±D2 ;
2) A2 p ± B q = 2CD.
                2
                                      142. x2 + y 2 = 13z 2 − 14z + 10. 143. 1) Îä-
íîïîëîñòíûé ãèïåðáîëîèä; 2) äâóïîëîñòíûé ãèïåðáîëîèä; 3) ãèïåð-
áîëè÷åñêèé ïàðàáîëîèä 3) ïàðàáîëè÷åñêèé öèëèíäð.                                   144. (1,1).
145. C(1, 1, −1); X 2 + Y 2 + Z 2 + 2XY − 2Y Z + 6XZ − 1 = 0.
146. 2x + y + 6 = 0. 147. 7x − 35y + 22 = 0, 7x + 14y + 20 = 0.
148. x−4y−2 = 0, x+4y−3 = 0. 149. x+y−1 = 0, 3x+3y+13 = 0.
150. z = 1; 2x − 3y = 0. 151. Ïðÿìàÿ öåíòðîâ 4x + 2y − 5 = 0.
152. Ïðÿìàÿ öåíòðîâ x = 1, y = t, z = −t, 4XY + 4XZ − 1 = 0.
153. X+Y +Z = 0. 154. 6x−2y+19 = 0, 2x+2y−1 = 0. 155. 7x+1 =
0. 156. 3x + 1 = 0, 3z − 2 = 0. 157. 7x + 17y + 19z + 19 = 0.
               2    2
158. Ýëëèïñ X9 + Y4 = 1; öåíòð C(2, 3), óãëîâîé êîýôôèöèåíò áîëü-
                                  2   2
øåé îñè − 21 . 159. Ãèïåðáîëà X4 − Y9 = 1; öåíòð C(1, 1), óãëîâîé
êîýôôèöèåíò äåéñòâèòåëüíîé îñè 32 . 160. Ïàðà äåéñòâèòåëüíûõ ïå-
ðåñåêàþùèõñÿ ïðÿìûõ x − y − 1 = 0, x − 4y + 2 = 0. 161. Ïàðàáîëà,
                          2    2                      2    2
X 2 = 2Y . 162. Ýëëèïñ X16 + Y4 = 1. 163. Ãèïåðáîëà X9 − Y25 = 1.


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