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Âûÿñíÿåì âçàèìíîå ðàñïîëîæåíèå íàïðàâëÿþùèõ âåêòîðîâ v1 è v2 . Åñëèv1 ||v2 , òî äàëåå èññëåäóåì íà ïàðàëëåëüíîñòü/ñîâïàäåíèå, èíà÷å ïåðåñå÷åíèå/ñêðåùèâàíèå.(a) Èòàê, ñ óðàâíåíèé ïðÿìûõ íàõîäèì v1 = [2, −2, −1], v2 = [−2, 3, 0],r1 = [1, 2, 0], r2 = [0, −5, 4], ∆r = [1, 7, −4].Âû÷èñëÿåì v1 × v2 = [3, 2, 2] 6= 0, ñëåäîâàòåëüíî, ïðÿìûå ëèáî ïåðåñåêàþòñÿ, ëèáî ñêðåùèâàþòñÿ. Âû÷èñëèì (v1 × v2 ) · ∆r = 9 6= 0, ïîýòîìóïðÿìûå ñêðåùèâàþòñÿ.(b) Äëÿ ðåøåíèÿ âòîðîé çàäà÷è íàéäåì íàïðàâëÿþùèå âåêòîðû è êàêèåíèáóäü òî÷êè íà äàííûõ ïðÿìûõ.
Íàõîäèì íàïðàâëÿþùèé âåêòîð ïåðâîéïðÿìîé v1 = [9, 5, 1] è òî÷êó íà íåé r1 = [0, 0, −3].Òàê êàê âòîðàÿ ïðÿìàÿ çàäàíà êàê ëèíèÿ ïåðåñå÷åíèÿ äâóõ ïëîêîñòåé,òî åå íàïðàâëÿþùèì âåêòîðîì áóäåò âåêòîðíîå ïðîèçâåäåíèå âåêòîðîâíîðìàëåé ê ïëîñêîñòÿì: n1 = [2, −3, −3], n2 = [1, −2, 1], ñëåäîâàòåëüíî,v2 = n1 × n2 = [−9, −5, −1]. Âèäèì, ÷òî v1 è v2 êîëëèíåàðíû, ñëåäîâàòåëüíî, ïðÿìûå ëèáî ïàðàëëåëüíû ëèáî ñîâïàäàþò. îáùåì ñëó÷àå äëÿ òîãî, ÷òîáû íàéòè òî÷êó íà ïðÿìîé, ðåøàåì ñèñòåìó óðàâíåíèé, ïîäñòàâëÿÿ âìåñòî îäíîé èç ïåðåìåííûõ êàêîå-íèáóäüäîñòàòî÷íî ¾õîðîøåå¿ ÷èñëî.
 äàííîé çàäà÷å çàìå÷àåì, ÷òî òî÷êà r1 =[0, 0, −3] óäîâëåòâîðÿåò óðàâíåíèþ âòîðîé ïðÿìîé. Ïîýòîìó ýòè ïðÿìûåñîâïàäàþò.JÏðèìåð 8 Äëÿ ïðÿìûõ x = 3 + t,y = 1 − t, z = 2 + 2tíàéäèòå:(à) êðàò÷àéøåå ðàññòîÿíèå ìåæäó íèìè;(b) óðàâíåíèå îáùåãî ïåðïåíäèêóëÿðà ê íèì.2 + 3t, z = 3tè x = −t,I Ñïèñûâàåì íåîáõîäèìûå âåêòîðû ñ óðàâíåíèé ýòèõ ïðÿìûõ:r1 = [3, 1, 2],, r2 = [0, 2, 0],v1 = [1, −1, 2],12v2 = [−1, 3, 3].y =Ðàññòîÿíèå ìåæäó ïðÿìûìè âû÷èñëÿåì ïî ôîðìóëå:d=|(v1 × v2 ) · ∆r|18.=√kv1 × v2 k110Óðàâíåíèå îáùåãî ïåðïåíäèêóëÿðà ìîæíî ïîëó÷èòü êàê ëèíèþ ïåðåñå÷åíèÿ ïëîñêîñòåé, ñîäåðæàùèõ êàæäóþ èç äàííûõ ïðÿìûõ è îáùèõ ïåðïåíäèêóëÿð.Íàõîäèì íàïðàâëÿþùèé âåêòîð îáùåãî ïåðïåíäèêóëÿðà v = v1 × v2 =[−9, −5, 2].Ñîñòàâëÿåì óðàâíåíèÿ ïëîñêîñòåé, ñîäåðæàùèõ îáùèé ïåðïåíäèêóëÿð:(v, v1 , r − r1 ) = 0 =⇒ 4x − 10y − 7z − 12 = 0,(v, v2 , r − r2 ) = 0 =⇒ 21x − 25y + 32z − 50 = 0.J4.Ïðåîáðàçîâàíèå êîîðäèíàòñòàðîéñòàðîìÏóñòü çàäàíû äâà áàçèñà (ðåïåðà): áàçèñ he1 , e2 , e3 i èñõîäíîé ()ñèñòåìû êîîðäèíàò è áàçèñ he01 , e02 , e03 i äðóãîé () ñèñòåìû êîîðäèíàò.Êàæäûé âåêòîðèìååò íåêîòîðûå êîîðäèíàòû â:ðåïåðåíîâîéíîâîãî ðåïåðàe01 = t11 e1 + t21 e2 + t31 e3 ,e02 = t12 e1 + t22 e2 + t32 e3 ,e03 = t13 e1 + t23 e2 + t33 e3 .Ìàòðèöåé ïåðåõîäà îò ñòàðîãî ðåïåðà he1, e2, e3i ê íîâîìó ðåïåðó he01, e02, e03iíàçûâàåòñÿ ìàòðèöà T , ñîñòàâëåííàÿ èç êîîðäèíàòíûõe0i , (i = 1, 2, 3), ò.å.
ìàòðèöà"#T =t11 t12 t13t21 t22 t23t31 t32 t33ñòîëáöîâ âåêòîðîâ.Ìàòðèöà ïåðåõîäà ìåæäó ðåïåðàìè íà ïëîñêîñòè îïðåäåëÿåòñÿ àíàëîãè÷íî.Åñëè ñòàðûé è íîâûé ðåïåðû îðòîíîðìèðîâàííûå è îäèíàêîâî îðèåíòèðîâàííûå, òî ìàòðèöà T ïåðåõîäà ìåæäó íèìè íàçûâàåòñÿ. Ìàòðèöàíà óãîë ϕ èìååò âèä:cos ϕ − sin ϕT =.ïîâîðîòàïîâîðîòà íà ïëîñêîñòèsin ϕ13cos ϕìàòðèöåéÔîðìóëû ïðåîáðàçîâàíèÿ êîîðäèíàò òî÷êèíà ïëîñêîñòè è â ïðîñòðàíñòâåÏðåîáðàçîâàíèåR2R3Ïðåîáðàçîâàíèå áàçèñíîãî Ñäâèã íà÷àëà ñ ñîðåïåðàñ ñîõðàíèåì íà÷àëà õðàíåíèåìðåïåðà êîîðäèíàòàõ ìàòðè÷íîé ôîðìåx = t11 x0 + t12 y 0 ,00 y = t21x + t22y .
0 xt tx= 11 12 · 0yyt21 t22x = x + x0 ,0 y = y + 0y0 . xxx= 0 + 0yy0y êîîðäèíàòàõ000x = t11 x + t12 y + t13 z ,y = t21 x0 + t22 y 0 + t23 z 0 ,z = t x0 + t32 y 0 + t33 z 0 ." # 31"# " 0#x = x + x0 , ìàòðè÷íîé ôîðìåxyz=t11 t12 t13t21 t22 t23t31 t32 t33x· y0z0y = y 0 + y0z = z 0 + z0 .xyz" 0#" #=Îáîçíà÷åíèÿ:xy0z0x0+ y0z0"#[x, y, z] êîîðäèíàòû íåêîòîðîé òî÷êè M â ñòàðîé ñèñòåìå êîîðäèíàò(ñòàðîì áàçèñå);[x0 , y 0 , z 0 ] êîîðäèíàòû òîé æå òî÷êè M â íîâîé ñèñòåìå êîîðäèíàò (íîâîìáàçèñå);[x0 , y0 , z0 ] êîîðäèíàòû íîâîãî íà÷àëà êîîðäèíàò O0 .Ïðèìåð 9 Äàíû äâå ñèñòåìû êîîðäèíàò Oxy è O x y .
Ïî îòíîøåíèþ0 0 0ê ïåðâîé ñèñòåìå êîîðäèíàò íà÷àëî âòîðîé íàõîäèòñÿ â òî÷êåO0 [−1, 3],0åäèíè÷íûå âåêòîðû âòîðîãî ðåïåðà èìåþò êîîðäèíàòû e1 = [2, 3], e02 =[1, 1]. Íàïèøèòå âûðàæåíèÿ êîîðäèíàò òî÷åê îòíîñèòåëüíî ïåðâîé ñèñòåìû êîîðäèíàò ÷åðåç èõ êîîðäèíàòû âî âòîðîé ñèñòåìå êîîðäèíàò.I Ïðåîáðàçîâàíèå êîîðäèíàò ÿâëÿåòñÿ ïîñëåäîâàòåëüíûì âûïîëíåíèåìäâóõ ïðåîáðàçîâàíèé: ïðåîáðàçîâàíèÿ ðåïåðà ñ ñîõðàíåíèåì íà÷àëà êîîðäèíàò è ñäâèãà ïîëó÷åííîãî ðåïåðà â íîâîå íà÷àëî. Ïîñëåäîâàòåëüíîåâûïîëíåíèå ýòèõ ïðåîáðàçîâàíèé (èõèëè) äàåò èñêîìîå ïðåîáðàçîâàíèå.
Ïðè ýòîì íåò ïðèíöèïèàëüíîé ðàçíèöû êàêîåïðåîáðàçîâàíèå âûïîëíèòü â ïåðâóþ î÷åðåäü, à êàêîå âî âòîðóþ. Ïîýòîìó íàéäåì ñíà÷àëà ïðåîáðàçîâàíèå ðåïåðà, à çàòåì ïåðåíåñåì åãî â íîâîåíà÷àëî.Ïîñêîëüêó èçâåñòíû êîîðäèíàòû âåêòîðîâ íîâîãî ðåïåðà, òî ñðàçó ñîñòàâëÿåì ìàòðèöó ïåðåõîäà:2 1T =.êîìïîçèöèÿ3141ñóïåðïîçèöèÿÏîëó÷àåì èñêîìîå ïðåîáðàçîâàíèå: 0 x2 1x−1=·+,0yèëè â êîîðäèíàòíîé ôîðìå31y3(x = 2x0 + y 0 − 1,y = 3x0 + y 0 + 3.JÏðèìåð 10 Íîâàÿ ñèñòåìà êîîðäèíàò ïîëó÷åíà èç ñòàðîé ñèñòåìû êîîðäèíàò ïåðåíîñîì12 íà÷àëà êîîðäèíàò â òî÷êó O0[3, −4] è ïîâîðîòîì íàóãîë α = − arccos 13 .
Íàéäèòå êîîðäèíàòû òî÷êè A[6, −2] â íîâîé ñèñòåìå êîîðäèíàò.12I Íàéäåì ìàòðèöó ïîâîðîòà ðåïåðà â ïëîñêîñòè äëÿ äàííîãî α = − arccos 13:112 5.T =13 −5 12Òîãäà ôîðìóëû ïðåîáðàçîâàíèÿ êîîðäèíàò èìåþò âèä:(1x = 13(12x0 + 5y 0 ) + 3,1y = 13(−5x0 + 12y 0 ) − 4.Èçâåñòíû ñòàðûå êîîðäèíàòû òî÷êè (x, y) = (6, −2).
Ïîýòîìó äëÿ òîãî,÷òîáû íàéòè êîîðäèíàòû (x0 , y 0 ) ýòîé æå òî÷êè â íîâîé ñèñòåìå êîîðäèíàò,íóæíî ðåøèòü ñèñòåìó óðàâíåíèé:((116 = 13(12x0 + 5y 0 ) + 3,(12x0 + 5y 0 ) = 3,⇐⇒ 13110000−2 = 13 (−5x + 12y ) − 4,13 (−5x + 12y ) = −2.Îòêóäà êîîðäèíàòû òî÷êè A â íîâîé ñèñòåìå (x0 , y 0 ) = (2, 3).5.JÏðèâåäåíèå îáùåãî óðàâíåíèÿ êðèâîé âòîðîãî ïîðÿäêà ê êàíîíè÷åñêîìó âèäóÎáùèì óðàâíåíèåì êðèâîé âòîðîãî ïîðÿäêà íàçûâàåòñÿ óðàâíåíèå âèäàAx2 + 2Bxy + Cy 2 + 2Dx + 2Ey + F = 0.Óäîáíî çàïèñûâàòü ýòî óðàâíåíèå â ìàòðè÷íîì âèäå X T ∆X = 0, ãäåA B ExAB∆ = B C D ,δ=,X = y .B C1E D FÎáùåå óðàâíåíèå êðèâîé îïðåäåëÿåò îäíó èç ñëåäóþùèõ ëèíèé:15(I) |δ| 6= 0. Ê ïåðâîé ãðóïïå îòíîñÿòñÿ ëèíèè, èìåþùèå åäèíñòâåííûéöåíòð ñèììåòðèè:1 ýëëèïñ,22xy+ 2 = 0 äâå ìíèìûå ïåðåñåêàþùèåñÿ ïðÿìûå,a2b−1 ìíèìûé ýëëèïñ,(221 ãèïåðáîëà,xy− 2 =2ab0 ïåðåñåêàþùèåñÿ ïðÿìûå.(II) |δ| = 0, |∆| 6= 0.
Êî âòîðîé ãðóïïå îòíîñÿòñÿ ëèíèè, íå èìåþùèåöåíòðà ñèììåòðèè:y 2 = 2px ïàðàáîëà.(III) |δ| = 0, |∆| = 0. Ê òðåòüåé ãðóïïå îòíîñÿòñÿ ëèíèè, èìåþùèå ïðÿìóþöåíòðîâ ñèììåòðèè:2 a , a 6= 0 äâå ïàðàëëåëüíûå ïðÿìûå,2x = −a2 , a 6= 0 äâå ìíèìûå ïàðàëëåëüíûå ïðÿìûå,0 äâå ñîâïàâøèå ïðÿìûå.öåíòðîì ëèíèèÒî÷êà (x0 , y0 ) íàçûâàåòñÿâòîðîãî ïîðÿäêà, åñëè åå êîîðäèíàòû óäîâëåòâîðÿþò ñèñòåìå óðàâíåíèé " x0 # A B D0· y0 =B C E01Ñóùåñòâóåò íåñêîëüêî ñïîñîáîâ ïðèâîäåíèÿ óðàâíåíèå êðèâîé ê êàíîíè÷åñêîìó. Èõ ïðèíöèïèàëüíîå îòëè÷èå ñîñòîèò â òîì, êàêîå ïðåîáðàçîâàíèå ïðîâîäèòü â ïåðâóþ î÷åðåäü ïîâîðîò èëè ïàðàëëåëüíûé ïåðåíîñ.Ðàññìîòðèì îáà ñïîñîáà.Ìåòîä ¾ïîâîðîò→ïåðåíîñ¿.Ïîâîðîò.Öåëü ïðèìåíåíèÿ ïðåîáðàçîâàíèÿ ïîâîðîòà èñêëþ÷èòü èçóðàâíåíèÿ ñëàãàåìîå, ñîäåðæàùåå ïðîèçâåäåíèå xy . Ïîýòîìó ïðåäïîëîæèì, ÷òî êîýôôèöèåíò B 6= 0.Èñêëþ÷èòü B èç óðàâíåíèÿ êðèâîé ìîæíî äâóìÿ ñïîñîáàìè.
Âíàõîäÿò óãîë ïîâîðîòà ϕ, ïðåîáðàçîâàíèå êîîðäèíàò è ýòèâûðàæåíèÿ ïîäñòàâëÿþò â èñõîäíîå óðàâíåíèå.ïåð-âîì ñëó÷àå16(1) Ïîâîðîò ñèñòåìû êîîðäèíàò çàäàåòñÿ ñèñòåìîé ðàâåíñòâ: xcos ϕ − sin ϕx1=·,ysin ϕcos ϕy1ãäå óãîë ϕ íàõîäèì òàê:(à) åñëè A 6= C , òî ϕ ÿâëÿåòñÿ ðåøåíèåì óðàâíåíèÿ:A−Ctg ϕ − 1 = 0;B(b) åñëè A = C , òî ìîæíî ïîâåðíóòü íà óãîë ϕ = π4 .2B cos 2ϕ + (C − A) sin 2ϕ = 0 ⇐⇒ tg2 ϕ +(2) Äåëàåì çàìåíó êîîðäèíàò â îáùåì óðàâíåíèè êðèâîé.
 ðåçóëüòàòåïðèõîäèì ê óðàâíåíèþA1 x21 + C1 y12 + 2D1 x1 + 2E1 y1 + F = 0.Êîýôôèöèåíòû A1 , C1 , D1 , E1 ìîæíî âû÷èñëèòü íåïîñðåäñòâåííîèëè íàéòè ïî ôîðìóëàì: A1 0A BTD1 E1 = D E · T,=TT,0 C1B Cãäå ìàòðèöà T îáîçíà÷àåò ìàòðèöó ïîâîðîòà.Âòîðîé ñïîñîáòðåáóåò çíàíèÿ äîïîëíèòåëüíîé òåîðèè: äèàãîíàëèçàöèè ñèììåòðè÷íîé ìàòðèöû, êîòîðûé áîëåå ïîäðîáíî áóäåò ðàññìîòðåíïîçæå.(1) Ñîñòàâëÿåìõàðàêòåðèñòè÷åñêîå óðàâíåíèådetA−λBBC −λ=0ñîáñòâåííûå ÷èñëà λ1 è λ2.(2) Íàõîäèì êîîðäèíàòû áàçèñíûõ (ñîáñòâåííûõ) âåêòîðîâ e0i êàê ðåøåè èùåì åãî êîðíè íèå ñèñòåìû óðàâíåíèé: A − λiBx0·=, i = 1, 2.BC − λiy0(3) Íîðìèðóåì íàéäåííûå âåêòîðû è èç èõ êîîðäèíàò ïî ñòîëáöàì ñîñòàâëÿåì ìàòðèöó ïîâîðîòà T .(4)  ðåçóëüòàòå ñðàçó çàïèñûâàåì A1 = λ1 è C1 = λ2 .
Âû÷èñëÿåì êîýôôôèöèåíòû D1 è E1 êàê â ïðåäûäóùåì âàðèàíòå ðåøåíèÿ è çàïèñûâàåì óðàâíåíèå â íîâûõ êîîðäèíàòàõ: A1 x21 + C1 y12 + 2D1 x1 +2E1 y1 + F = 0.17Ïàðàëëåëüíûé ïåðåíîñ.ñïîñîáîâ ñîâïàäàþò.Äàëåå õîä ðåøåíèÿ äëÿ ïåðâîãî è âòîðîãî(1) (a) Åñëè A1 C1 6= 0, òî â óðàâíåíèèA1 x21 + C1 y12 + 2D1 x1 + 2E1 y1 + F = 0ïðèñóòñòâóþò îáà êâàäðàòà ïåðåìåííûõ x è y .
Âûäåëÿåì ïîëíûåêâàäðàòû ñ x è y :A1 (x1 − x0 )2 + C1 (y1 − y0 )2 + F1 = 0.(b) Åñëè îäèí èç êîýôôèöèåíòîâ ïðè êâàäðàòàõ ðàâåí íóëþ, ïóñòü,äëÿ îïðåäåëåííîñòè, A1 = 0, òî óðàâíåíèå C1 y12 +2D1 x1 +2E1 y1 +F = 0 ïðèâîäèòñÿ ê âèäóC1 (y1 − y0 )2 + 2E1 (x1 − x0 ) = 0.(2) Ïåðåîáîçíà÷àåì ïåðåìåííûå: xK = x1 −x0 , yK = y1 −y0 è çàïèñûâàåìêàíîíè÷åñêîå óðàâíåíèå êðèâîé22A1 x2K + C1 yK+ F1 = 0 èëè C1 yK+ 2E1 xK = 0.(3) Íàõîäèì âûðàæåíèå ñòàðûõ êîîðäèíàò ÷åðåç êàíîíè÷åñêèå: xcos ϕ − sin ϕxK + x0=·,ysin ϕcos ϕyK + y0ãäå e01 = [cos ϕ, sin ϕ]T , e02 = [− sin ϕ, cos ϕ]T è O0 (x0 , y0 ) åñòü âåêòîðûêàíîíè÷åñêîãî áàçèñà è íîâîå íà÷àëî êîîðäèíàò.Ìåòîä ¾ïåðåíîñ→ïîâîðîò¿Åñëè |δ| 6= 0, òî äëÿ óïðîùåíèÿ ïðåîáðàçîâàíèé ìîæíî ïåðåíåñòè íà÷àëî êîîðäèíàò â öåíòð ëèíèè âòîðîãî ïîðÿäêà, ò.å.
âûïîëíèòü. Íàõîäèì:ïàðàë-ëåëüíûé ïåðåíîñ(a) öåíòð ëèíèè âòîðîãî ïîðÿäêà (x0 , y0 );(b) ñäâèã ñèñòåìû êîîðäèíàò:x = xc + x0 ,18y = yc + y 0 ;(c) çàìåíÿåì êîîðäèíàòû â óðàâíåíèè:Ax2c + 2Bxc yc + Cyc2 + F1 = 0,ãäå F1 = Dx0 + Ey0 + F .Ïîâîðîò îñóøåñòâëÿåòñÿ îäíèì èç îïèñàííûõ âûøå ñïîñîáîâ.Åñëè |δ| = 0, òî ïðèâåäåíèå êðèâîé âòîðîãî ïîðÿäêà ê êàíîíè÷åñêîìóâèäó îñóùåñòâëÿåòñÿ ïî ñõåìå ¾ïîâîðîò → ïàðàëåëüíûé ïåðåíîñ¿.Ïðèìåð 11 Îïðåäåëèòå âèä è ðàñïîëîæåíèå ëèíèè âòîðîãî ïîðÿäêà, çàäàííîé óðàâíåíèåì:(a) 7x22 + 6xy − y2 +2 14x + 6y + 23 = 0;(b) 9x + 12xy + 4y − 16x − 2y + 6 = 0.I (a)I ñïîñîá. Èç óðàâíåíèÿtg2 ϕ +8tg ϕ − 1 = 0.3íàõîäèì òàíãåíñ óãëà ïîâîðîòà tg ϕ = 13 èëè tg ϕ = −3.
Âûáåðåì ïîëîæèòåëüíîå çíà÷åíèå òàíãåíñà, ñîîòâåòñòâóþùåå ïîâîðîòó ñèñòåìû êîîðäèíàòïðîòèâ ÷àñîâîé ñòðåëêè. Åñëè â ðåçóëüòàòå ýòîãî ïðåîáðàçîâàíèÿ èçìåíèòñÿ ïîðÿäîê êîîðäèíàò, òî âûïîëíèì åùå îäíî ïðåîáðàçîâàíèå, äîïîëíèòåëüíî ïîâåðíóâ íà óãîë 90◦ èëè ïåðåèìåíîâàâ îñè.Èòàê, ïóñòü tg ϕ = 31 =⇒ cos ϕ = √310 , sin ϕ = √110 . Ìàòðèöà ïîâîðîòàáóäåò èìåòü âèä √√T =à ïðåîáðàçîâàíèå êîîðäèíàò:(x=y=3/√101/ 10−1/√103/ 10,√1 (3x1 − y1 ),10√1 (x1 + 3y1 ).10Ïîäñòàíîâêó íàéäåííûõ âûðàæåíèé äëÿ ñòàðûõ êîîðäèíàò ìîæíî çàìåíèòü âû÷èñëåíèåì íåñêîëüêèõ ïðîèçâåäåíèé ìàòðèö: 11A103 1733 −180=√··√=,0 C13 −130 −210 −1 310 1113 −1[ D1 E1 ] = [ 7 3 ] · √= √ [ 24 2 ] .310 11019Òîãäà óðàâíåíèå ëèíèè â íîâûõ êîîðäèíàòàõ ïðèìåò âèä:4848x21 − 2y12 + √ x1 + √ y1 + 23 = 0.1010Äàëåå ãðóïïèðóåì ñëàãàåìûå ñ îäèíàêîâûìè ïåðåìåííûìè è äîïîëíÿåìäî êâàäðàòîâ:69722128 x21 + √ x1 +−− 2 y12 − y1 +++ 28 = 0;1010101010102213− 2 y1 − √+ 16 = 0.8 x1 + √1010Ââîäèì íîâûå êîîðäèíàòû x2 = x1 + √310 , y2 = y1 − √110 .  ðåçóëüòàòåïîëó÷àåì ¾ïî÷òè êàíîíè÷åñêîå¿ óðàâíåíèå ãèïåðáîëû:8x22 − 2y22 = −16.Äëÿ òîãî, ÷òîáû åãî îêîí÷àòåëüíî ïðèâåñòè ê êàíîíè÷åñêîìó âèäó, ïåðåèìåíóåì êîîðäèíàòíûå îñè: xK = y2 , yK = x2 , è ðàçäåëèì îáå åãî ÷àñòèíà −16.
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