TIInterActive File'Il moto circolare uniforme, la circonferenza, l'ellisse e l'iperbole nel piano cartesiano (si prevedono circa 26 ore di lavoro in classe) Prerequisiti: funzioni lineari, successioni lineari; concetto di funzione; differenze finite; polinomi e operazioni con essi; il moto rettilineo uniforme e la retta nel piano cartesiano; il moto uniformemente accelerato e la parabola nel piano cartesiano. Premessa Ci occupiamo, in questa lezione, del moto circolare uniforme, che ci porter a trattare la circonferenza nel piano cartesiano. Vedrmo poi come sia possibile ricavare, a partire dall'equazione di una circonferenza, quella di un'ellisse avente gli assi di simmetria paralleli agli assi cartesiani. Infine introdurremo lo studio analitico dell'iperbole come luogo di punti. Scheda 1. Moto circolare uniforme (Si prevedono circa 2 ore di lavoro in classe) Fra i moti curvilinei assume particolare importanza quello di un oggetto che si muove su una circonferenza con velocit in modulo costante: tale moto si dice circolare uniforme. Nel moto circolare uniforme, l'accelerazione sempre diretta verso il centro della circonferenza descritta dal corpo in moto. Ci vuol dire che, nella direzione perpendicolare, ossia lungo la tangente alla traiettoria, la componente dell'accelerazione nulla. Ecco perch nel moto circolare uniforme la velocit (che diretta lungo la tangente alla circonferenza descritta dal corpo) costante in modulo: la velocit varia in direzione e verso (e ci giustifica la presenza di un'accelerazione), ma costante in modulo. Per la descrizione di un punto che si muove di moto circolare uniforme sono importanti le seguenti grandezze: a) raggio r della circonferenza; b) periodo, indicato in genere con T: tempo impiegato dal corpo a percorrere un giro completo (ossia la circonferenza); c) frequenza, indicata in genere con f: numero di giri percorsi nell'unit di tempo (si misura in Hertz); d) velocit angolare, indicata in genere con w: rapporto fra l'angolo descritto dal punto e tempo impiegato a percorrerlo; e) velocit lineare, indicata in genere con v: rapporto tra arco percorso dal punto e tempo impiegato a percorrerlo; La velocit angolare legata alla velocit lineare dalla relazione: v = w * r L'accelerazione centripeta legata alla velocit lineare dalla relazione: a = v^2 / r e alla velocit angolare dalla relazione: a = w^2*r Attivit 1. (a coppie o anche individualmente) Leggete con attenzione le pagine sul moto circolare uniforme accessibili cliccando sull'hotword moto circolare uniforme. Eventuali parti che non abbiamo capito o domande che vogliamo porre all'insegnante Attivit 2 (in laboratorio in piccoli gruppi) Come possibile misurare l'accelerazione di gravit usando un giradischi come cronometro? Provate voi stessi a produrre un piano di lavoro e poi confrontatelo con quello presente sul sito intrnet accessibile cliccando l'hotword esperimento . Provate quindi ad eseguire voi stessi l'esperimento che viene suggerito. Il risultato dell'esperimento con breve riferimento a eventuali difficolt incontrate, teoriche o pratiche Problemi (da risolvere individualmente) 1. Un satellite in unorbita circolare di raggio 220km sopra la superficie terrestre effettua una rivoluzione completa in 89 minuti. Calcola: a) La velocit v del satellite b) Laccelerazione a del satellite 2. La luna ruota intorno alla Terra ad una distanza di 3.84*105km. Il periodo di circa 28giorni. Qual l a sua accelerazione? La mia risposta Scheda 2. Il moto circolare uniforme, l'equazione di una circonferenza e relativi problemi (Si prevedono circa 8 ore di lavoro in classe) Nel caso del moto di un proiettile avete visto come sia semplice ricavare l'equazione dela traiettoria a partire dalla conoscenza delle leggi orarie lungo x e y. Anche il moto circolare uniforme un moto piano e quindi dovrebbe essere possibile seguire un procedimento simile a quello visto per il moto del proiettile per ricavare l'equazione della traiettoria, ossia l'equazione cartesiana di una circonferenza: a) determinare come variano, rispetto al tempo, l'ascissa e l'ordinata di un punto che si muove di moto circolare uniforme; b) ricavre il tempo da una delle due equazioni, in modo da ottenere una relaizone tra x e y del tipo f(x; y)=0, ossia l'equazione della traiettoria. Il problema che, nel caso della circonferenza, queste operazioni non sono semplici come nel caso della parabola, nel senso che richiedono la conoscenza du due funzioni che non avete ancora studiato e di cui ci occuperemo nella prossima lezione. Invece molto semplice determinare l'equazione cartesiana di una circonferenza a partire dalla sua definzione come luogo di punti equidistanti dal centro. Prima di passare a determinare l'equazione cartesiana di una circonferenza di centro C(x0; y0) e raggio r, vi chiediamo di svolgere la seguente attivit: Attivit 3 (in piccoli gruppi) Cercate di descrivere, qualitativamente, le principali caratteristiche della variazione, rispetto al tempo, dell'ascissa e dell'ordinata di un punto che si muove di moto circolare uniforme su una circonferenza di centro C(x0; y0) e raggio r . Confrontate la vostra descrizione con quella di altri gruppi. La nostra descrizione La nostra descrizione dopo il confronto con altri gruppi Attivit 4 (in piccoli gruppi) La circonferenza l'insieme di tutti e soli i punti che sono equidistanti da un punto detto centro. La distanza di un qualunque punto della circonferenza dal centro si dice raggio. Dato un centro C e un raggio r, dimostrate che l'equazione della circonferenza di centro C e raggio r, scegliendo un sistema di riferimento cartesiano avente l'origine coincidente con C : x^2 + y^2 = r^2 La nostra dismostrazione Attivit 5 (in piccoli gruppi) Dimostrate che l'equazione di una circonferenza di centro C(x0; y0) e raggio r (x - x0)^2 + (y - y0)^2 = r ^2 La nostra risposta Attivit 6 (individuale) Clicca sull'hotword seguente; leggi attentamente quanto scritto e completa l'esercitazione attivit 6 La mia risposta alle domande ed eventuali punti non chiari Esercizi di applicazione sulla circonferenza (da svolgere individualmente) Determina centro e raggio delle circonferenze che hanno le seguenti equazioni e poi disegnale: x2 + y2 - 2x - 4y - 4 = 0 2x2 + 2y2- 2x - 6y - 3 = 0 x2 +y2 - 4y = 0 5x2 +5y2- 4x = 0 2. Qual il luogo dei punti che hanno da (1,1) distanza tripla che da (5,1)? 3. Disegna la circonferenza di equazione x2 +y2 +6x - 2y +1 = 0. In quali punti interseca lasse x? 4. Lequazione x2 +y2 = 1 rappresenta la circonferenza di centro (0,0) e raggio 1. Che luogo rappresenta la disequazione x2 +y2 > 1? E la disequazione x2 +y2 < 1? 5. Determina l'equazione della circonferenza che ha centro in (2,4) e raggio 2. 6. Determina l'equazione della circonferenza che passa per (0,0), (1,2), (2,3). 7. Determina lequazione della circonferenza che passa per i punti (1,1), (2,1), (0,3). 8.Determina i punti di intersezione tra la circonferenza dellesercizio precedente e la retta di equazione y = 2x+3. 9. Determina gli eventuali punti di intersezione tra le due circonferenze di equazione x^2+y^2 = 5 (x - 1)^2 + (y - 1)^2 = 3 10. Scrivi una procedura per determinare gli eventuali punti di intersezione tra due circonferenze. Le mie risposte Attivit 7 (in piccoli gruppi) Abbiamo visto che per imporre la condizione di tangenza tra una retta y = p*x+q e una parabola y = a*x^2 + b*x + c possibile determinare l'equazione risolvente il sistema tra l'equazione della retta e quella della parabola e imporre che il discriminante (il D) di questa equazione sia uguale a 0. A vostro parere possibile fare qualcosa di simile anche con la tangenza fra una retta e una circonferenza? Giustificate la risposta. La nostra risposta La nostra risposta dopo il confronto con altri gruppi di lavoro Attivit 8 (in piccoli gruppi) Nel caso di tangenza fra una retta e una circonferenza vale il seguente teorema: la retta tangente a una circonferenza in un punto P e il raggio passante per P sono fra loro perpendicolari. Ci vuol dire che la distanza della retta tangente dal centro uguale al raggio. Come potreste utilizzare questo teorema per determinare, fra le infinite rette del fascio y = x + q quelle tangenti alla circonferenza di equazione x^2 + y^2 = 5? Ottenete lo stesso risultato imponendo che il discriminante (il D) dell'equazione risolvente il sistema fra l'equazione della retta e quella della circonferenza sia nullo? La nostra risposta La nostra risposta dopo il confronto con altri gruppi Attivit 9 (in piccoli gruppi) Cercate di scrivere due procedure per la determinazione della condizione di tangenza fra una retta e una circonferenza, una basata sull'imporre uguale a 0 il discriminante dell'equazione risolvente e un'altra fondata sul teorema che afferma la perpendicolarit, nel punto di tangenza, tra tangente e raggio. La nostra risposta La nostra risposta dopo il confronto con altri gruppi La nostra risposta dopo l'eventuale sistemazione dell'insegnante Problema (da svolgere in piccoli gruppi) Determinate le equazioni delle circonferenze aventi i centri appartenenti alla retta di equazione y = 4*x+1 e tangenti a entrambi gli assi (svolgete l'esericizio in due modi diversi) La nostra risposta e il confronto fra le due risoluzioni (eventualmente dopo il suggerimento d altri gruppi di lavoro o dell'insegnante) Attivit 10 (a coppie) Una circonferenza non si pu esprimere nella forma y = f(x), questo perch la circonferenza non pu essere il grafico di una funzione: infatti una funzione assegna ad ogni x una sola y. La circonferenza, per, si pu vedere come lunione di due funzioni: la semicirconferenza che sta sopra e la semicirconferenza che sta sotto. In questo modo possibile tracciare una circonferenza anche nell'ambiente grafico di TI-InterActive!; vediamo per esempio come si rappresenta la circonferenza di centro (1, 1) e raggio 2: lequazione : [P\P^0PPPPPPvP!P1@P2@P#PP PPPY PZ P_ Pj P  P SP_ Pu P $P @P   L@P rPR PZ Pz @P @P !@P< @P> @PL PM >P dP AP @P P ZPA@PB@P @P@P@P>@PP PZP0@PN@PO@PR@PSM@PPPP9PP P PP P4P@P@P@P!@P"@P2P8P9 PEPFPUPo PykP PAP4-PaP`@P<@P@P<@P@P@P@P@P@P@P @P<@P@P@P<@P@P@P@P@P @P @P @P @P@P@P@P<@P@P@P<@P@P@P@P @P!@P'@P(<@P)@P,@P-<@P.@P0@P2@P3|@P@P<@P@P@P<@P@P@P@P@P@P@P)@P@P@P@P<@P@P@P<@Pd@Pc@Pd<@Pe@Pg@Ph<@Pi@P@P<@P@P@P<@Po@P@P@P@P@PP@PPP.P/PYP7 PA PJ!PK!lP!UP " P"MPc#P#P#P$Pc% Po% Pz%P{%3@P%@P%@P%@P%r@P'&@P(& @P2&@P3&a@Pp"Arial"Arial"Arial"ArialSymbol"Arial"Arial"Verdana[\\]]^   1223ab"#)*  * + T U ^ _ _ `    ^ _ _ ` ` a a b b c c d d e e f f g g h h i i j j k k l l m m n n o o p p q q r r s s t t u     M N N O O P P Q Q R y z z {  ( ) K L L M           ABBCqr !!"NOOPno12RSSTTUghhiijjkkllmmnno-..//001122334~%&78./01YZef     )*UV34455667UV  !!!!!!!!!!!!!!!!""""""""" " " " " " " "+",","-"a#b#b#c#u#v#v#w#w#x#x#y#y#z#################$ $$$$$`%a%a%b%b%c%y%z%z%{%'''dL  \ l0!#dL  \ l0!#d p@ P !$`'d p@ P !$`'dp@ P !$`'Hhttp://www.na.infn.it/cpdf/Elaborati_Finali/Berr_Mas/moto_circolare.htmLhttp://www.na.infn.it/cpdf/Elaborati_Finali/Berr_Mas/lab_moto_circolare.htmJhttp://www.matematica.it\paola\Terzoanno\Geometriaanalitica\Attivita6.docArialArialArialArialSymbolArialArialVerdana7(x - 1)2+(y - 1)2 = 4 Fate risolvere quest'equazione a TI -InterActive! rispetto alla variabile y, in modo da ottenere le due semicirconferenze e poi fate disegnare le due semicirconferenze, per ottenere il grafico della circonferenza iniziale. La nostra risposta Scheda 3. Dalla circonferenza all'ellisse e relativi problemi (Si prevedono circa 8 ore di lvoro in classe) Nella precedente attivit avete visto come disegnare in TI- InterActive! la circonferenza di equazione (x - 1)2+(y - 1)2 = 4 . Ecco il disegno nella finestra [-8; 8] x [-8; 8] e le espressioni analitiche delle due semicirconferenze calcolate con TI-InterActive!   Notate che le due semicirconferenze sembrano non raccordarsi bene ... sapreste spiegare perch? La nostra risposta Attivit 11 (in piccoli gruppi) Provate a effettuare ingrandimenti e riduzioni della finestra [-8; 8] x [-8; 8] sopra scelta utilizzando uno stesso fattore di ingrandimento o di riduzione sia per l'asse x che per l'asse y. Che cosa ottenete? Ora cambiate il fattore, per esempio lasciando a 2 quello sull'asse x e scegliendo 1 sull'asse y. Che cosa vi aspettate accada? Provate a verificare la correttezza delle vostre congetture. Ora fate il viceversa: mettete a 2 quello sull'asse y e a 1 quello sull'asse x; che cosa vi aspettate accada? Provate a verificare la correttezza delle vostre congetture. La nostre congetture, quello che abbiamo osservato e le nostre consideraizoni finali Scegliere un fattore di zoom maggiore di 1 sull'asse delle x, vuol dire effettuare un ingrandimento lungo quest'asse, ossia stirare il piano cartesiano e quindi il grafico lungo l'asse x. Analogamente per quello che riguarda l'asse y. Quindi scegliere la coppia (2; 1) di ingrandimenti equivale a stirare i grafici contenuti nel piano cartesiano di un fattore 2 lung l'asse x; scegliere la coppia (2; 3) equivale a stirare i grafici di 2 lungo l'asse x e di 3 lungo l'asse y; il risultato finale quindi uno stiramento di 3/2 lungo l'asse verticale. Come possibile rappresentare analiticamente la coppia (a; b) di fattori di zoom (ossia un fattore a lungo x e un fattore b lungo y) centrati nell'origine? Il sistema di equazioni: X = x/a Y = y/b consente di determinare immediatamente la finestra risultante dopo uno zoom di coppia (a; b) centrato nell'origine. Infatti, mediante uno zoom di fattori (a; b) centrato nell'origine si passa dalla finestra [xMIN; xMAX] x [yMIN; yMAX] alla finestra [xMIN/a; xMAX/a] x [yMIN/b; yMAX/b] come potete verificare direttamente. Attivit 12 (in piccoli gruppi) Che cosa accade se lo zoom non centrato in (0;0). Per rispondere provate a centrare lo zoom in (1; 1) scegliendo, nel seguente grafico, una finestra centrata in (1;1) (per esempio [-8; 10] x [-8; 10] )  La nostra risposta D'ora innanzi parleremo di cambiamento di scala di fattori (a; b) per intendere una trasformazione che contrae o dilata i grafici di a lungo x e di b lungo y. La curva che si ottiene applicando a una circonferenza un cambiamento di scala di fattori (a; b) si dice ellisse. Attivit 13 (a coppie) Considerate la circonferenza di centro l'origine e raggio 2. Apllicate a essa un cambiamento di scala di fattori (2; 2). Che cosa ottenete? Applicate ora alla circonferenza di centro l'origine e raggio 2 un cambiamento di scala di fattori (2; 1). Che cosa ottenete? Applicate ora alla circonferenza di centro l'origine e raggio 2 un cambiamento di scala di fattori (3; 2). Che cosa ottenete? Applicate infine alla circonferenza di centro l'origine e raggio 2 un cambiamento di scala di fattori (1/2; 1/4). Che cosa ottenete? Rispondete a ciascuna domanda fornendo il grafico, l'equazione e le principali caratteristiche delle curve che ottenete. Le nostre risposte Attivit 14 (a coppie) Come si trasforma l'equazione di una circonferenza del tipo x^2 + y^2 = 1 applicando a essa un cambiamento di scala di fattori (a; b)? E una circonferenza del tipo (x - x0)^2 + (y - y0)^2 = 1? Le nostre risposte Effettuando la precedente attivit dovreste aver visto che una curva avente equazione x^2/a^2 + y^2/b^2 = 1 un'ellisse ottenuta applicando a una circonferenza di centro l'origine e raggio 1 un cambiamento di scala di fattori (a; b). Vogliamo ora far vedere che, con un'opportuna scelta del sistema di riferimento cartesiano, si ottiene un'equazione dello stesso tipo per il luogo geometrico dei punti per cui costante la somma delle distanze da due punti fissi detti fuochi. Attivit 15 (in piccoli gruppi) Considerate la seguente definizione: "Si dice ellisse il luogo geometrico dei punti del piano per cui costante la somma delle distanze da due punti fissi detti fuochi" Scegliete un sistema di riferimento cartesiano tale che l'asse delle x passi per i due fuochi F1 e F2 e l'asse delle y sia asse del segmento F1F2. Dimostrate che, in tale sistema di riferimento, l'ellisse di fuochi F1 e F2 ha un'equazione del tipo  La nostra risposta (eventuali problemi incontrati e non risolti) Cos come esistono il compasso per tracciare circonferenze, il meccanismo di Paucellier per tracciare rette, parabolografi per tracciare parabole, possibile costruire macchine matematiche che traccino ellissi. Uno di questi ellissografi simulato nei seguenti file di cabri II plus e nel corrispondente file di cabrijava. ellissografoplus ellissografojava Attivit 16 (in piccoli gruppi) Enunciate qualche propriet del grafico di un'ellisse del tipo  e dimostratele La nostra risposta La nostra risposta dopo il confronto con altri gruppi ed, eventualmente, con l'insegnante Per il prosieguo, precisiamo che si dice: a) asse maggiore dell'ellisse, il segmento intersecato sull'ellisse dalla retta passante per i fuochi. Nel caso dell'ellisse di equazione  avente i fuochi sull'asse x, l'asse maggiore ha lunghezza 2a; b) asse minore dell'ellisse, il segmento interscato sull'ellisse dalla perpendicolare all'asse maggiore passante per il centro dell'ellisse. Nel caso dell'ellisse di equazione  avente i fuochi sull'asse x, l'asse minore ha lunghezza 2b; c) vertici dell'ellisse, i punti di interesezione fra l'ellisse e i suoi assi; d) asse focale dell'ellisse, la distanza tra i due fuochi; e) "eccentricit" di un'ellisse, il rapporto tra il semiasse focale e il semiasse maggiore. L'eccentricit, che un numero compreso fra 0 e 1, indica una misura dello "schiacciamento" di un'ellisse: se l'eccentricit 1, l'ellisse degenera in un segmento; se l'eccentricit 0, l'ellisse una circonferenza. Attivit 17 (in piccoli gruppi) Ricavate, in funzione di a e di b le coordinate dei fuochi di un'ellisse del tipo  La nostra risposta Attivit 18 (in piccoli gruppi) Che cosa cambia nell'ellisse e nelle sue propriet se i fuochi stanno sull'asse y e non sull'asse x? La nostra risposta Attivit 19 (a coppie) Scrivete una procedura che consenta di determinare la condizione di tangenza fra una retta di equazione y = p*x + q e un'ellisse di equazione  . La nostra risposta Attivit 20 (a coppie) Che cosa cambia nell'equazione di un'ellisse che ha il centro in un punto C(h;k) invece che in O(0;0), ma che mantiene gli assi paralleli agli assi cartesiani? La nostra risposta Esercizi sull'ellisse (da svolgere individualmente) 1. Determina lequazione del luogo dei punti del piano che hanno da F1(-4,0) e da F2(4,0) somma delle distanze uguale a 10. 2. Disegna lellisse di equazione 4 x2 + 25y2 100 = 0. 3. Determina le coordinate dei fuochi e i semiassi delle seguenti ellissi e poi disegnale: 9 x2 + 36y2 = 324 8(x 1)2 + 9(y + 2)2 = 72 4. Determina lequazione dellellisse di centro O (che ha per assi gli assi x e y) che ha semiasse maggiore a=1 e eccentricit e=1/2 . 5. Determina i punti di intersezione tra lellisse x2/81 + y2/16 = 1 e la circonferenza di centro O e raggio r=6. 6. Scrivi lequazione di una ellisse di eccentricit e=0.8 e poi disegnala. Le mie risposte Scheda 4. L'iperbole e relativi problemi (Si prevedono circa 8 ore di lavoro in classe) Attivit 21 (in piccoli gruppi) Considerate l'equazione  e fate un piccolissimo cambiamento, mettendo al posto del segno "+" un segno "-". Come cambiano le caratteristiche della curva che rappresenta sul piano cartesiano l'insieme delle soluzioni dell'equazione  rispetto alle caratteristiche della curva che rappresenta  ? Suggerimenti per la risposta: a) le simmetrie vengono mantenute? b) Quali sono i punti di interesezione con gli assi cartesiani? c) Fate risolvere l'equazione rispetto a y:  e poi chiedetevi come tende a comportarsi la curva quando x diventa molto grande o molto piccolo e a quali parti di piano appartiene la curva; d) dove si trova l'ellisse di equazione  rispetto alla curva di equazione  ? La nostra risposta La nostra risposta dopo il confronto con altri gruppi La nostra risposta dopo l'intervento dell'insegnante Domanda 1 (individuale) Qui sotto abbiamo fatto disegnare a TI-InterActive! la curva di equazione  con le due rette di equazione y=3x/2 e y = -3x/2  Come cambierebbe il grafico passando dalla curva di equazione  alla curva di equazione  ? Prima di farlo disegnare a TI-InterActive!, prova a disegnare su un foglio il grafico che ti aspetti. La mia risposta (spiegazioni di eventuali differenze tra il grafico che ti attendevi e quello disegnato da TI -InterActive!) Domanda 2 (individuale) Come cambia il grafico passando dalla curva di equazione  alla curva di equazione  ? La mia risposta Le curve di equazione  o  si chiamano iperboli. Le rette di equazione y = b*x/a e y = -b*x/a , che delimitano le regioni di piano in cui l'iperbole situata, si chiamano asintoti dell'iperbole. L'iperbole viene definita come luogo geometrico di punti nel seguente modo: si dice iperbole il luogo geometrico di tutti e soli i punti del piano per cui costante il modulo della differenza delle distanze da due punti fissi detti fuochi. Attivit 22 (in piccoli gruppi) Siano dati due punti F1 e F2. Scegliete un sistema di riferimento cartesiano xOy tale che l'asse x passi per F1 e F2 e l'asse y sia asse del segmento F1F2. A partire dalla definizione di iperbole come luogo geometrico di punti, dimostrate che l'equazione di un'iperbole nel sistema di riferimento scelto del tipo  . Che cosa cambierebbe nell'equazione se aveste scelto i fuochi sull'asse y e l'asse delle x asse del segmento F1F2? Le nostre risposte Cos come esistono il compasso per tracciare circonferenze, il meccanismo di Paucellier per tracciare rette, parabolografi per tracciare parabole, ellissografi per tracciare ellissi, possibile costruire macchine matematiche che traccino iperboli. Uno di questi iperbolografi simulato nei seguenti file di cabri II plus e nel corrispondente file di cabrijava. iperbolografoplus iperbolografojava Attivit 23 (in piccoli gruppi) Ricavate, in funzione di a e di b le coordinate dei fuochi di un'iperbole del tipo  e quelle di un'iperbole del tipo:  La nostra risposta Attivit 24 (a coppie) Scrivete una procedura che consenta di determinare la condizione di tangenza fra una retta di equazione y = p*x + q e un'iperbole di equazione  . La nostra risposta Attivit 25 (a coppie) Che cosa cambia nell'equazione di un'iperbole del tipo  il cui il centro viene traslato da O(0;0) a un punto C(h;k)? La nostra risposta Un'iperbole per cui a=b si dice equilatera. Gli asintoti di un'iperbole equilatera sono quindi la bisettrici del primo e terzo quadrante (y=x) e la bisettrice del secondo e q uarto quadrante (y = -x). L'equazione di un'iperbole equilatera pu quindi essere del tipo  o anche del tipo  . Mediante una rotazione di 45 di un'iperbole equilatera, per possibile ottenere una curva i cui asintoti sono gli assi cartesiani e non pi le bisettrici dei quadranti. Naturalmente la curva che si ottiene sar sempre la setssa iperbole equilatera, ma la sua equazione cambier. In effetti, ruotando di 45 in senso antiorario una curva del tipo  si ottiene una curva che ha equazione x*y = k oppure, equivalentemente, . Agendo sul cursore puoi osservare l'effetto che il cambiamento di valore di k ha sui grafici di funzioni del tipo y=k/x.  Attivit 26 (in piccoli gruppi) Come cambia l'equazione di una curva del tipo  se la si trasla di un vettore di componenti (m; n)? La nostra risposta Attivit 27 (in piccoli gruppi) Disegnate la curva di equazione  e poi scrivete una procedura per disegnare una qualunque curva di equazione  . Esercizi sull'ellisse (da svolgere individualmente) 1. Determina lequazione del luogo dei punti del piano che hanno da F1(-4,0) e da F2(4,0) differenza delle distanze uguale a 3. 2. Disegna liperbole di equazione 4 x2 25y2 100 = 0. 3. Determina le coordinate dei fuochi e i vertici delle seguenti iperboli e poi disegnale: 9 x2 36y2 = 324 8(x 1)2 9(y + 2)2 = -72 4. Determina i punti di intersezione tra liperbole di equazione x2/81 y2/16 = 1 e la circonferenza di centro O e raggio r=6. Le mie risposte Prima di eseguire il solito test si autovalutazione, gli esercizi di consolidamento e le prove di verifica, ti chiediamo di fare tu, insieme a qualche compagno di gruppo, il documento di sintesi teorica di questa lezione. Confrontalo poi con quello proposto da altri compagni di gruppo e, infine, proponete la vostra sintesi all'insegnante. La nostra sintesi Un interessante lavoro sulle sezioni coniche, nel quale puoi ritrovare molte delle nozioni e tecniche apprese, approfondirne e conoscerne altre, stato effettuato dagli studenti dell'ITC Rondani di Parma. A esso puoi accedere cliccando sulla seguente hotword (non ti chiediamo una vera e propria attivit sugli argomenti presentati, ma solo una lettura che possa aiutarti a sistemare e a avedere da una diversa prospettiva alcuni argimenti che hai studiato) le coniche test di autovalutazione esercizi di consolidamento esempi di prove di verifica f /%PPP<P P PP<PPP=PR@PS@PPPP<PPPP<P(P"<P# P,<P-[Ph@PPP PTPf<Pg Pp<PqPcUPBP P P <P P <P P P P P% PPP PPP PP PPP PPPPP)P P P P kPw@PPF@PP"#P&67P? 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