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7. Ïðÿì³ õ = ± à/ε íàçèâàþòüñÿ äèðåêòðèñàìè ã³ïåðáîëè.<br />
Âîíè ìàþòü ò³ ñàì³ âëàñòèâîñò³, ùî ³ äèðåêòðèñè åë³ïñà.<br />
гâíÿííÿ ïàðàáîëè ó = àõ 2 , ÿêå ðîçãëÿíóòå â øê³ëüíîìó<br />
êóðñ³ ìàòåìàòèêè, ìîæíà îäåðæàòè ÿê ãåîìåòðè÷íå ì³ñöå<br />
æ 1 ö<br />
òî÷îê, ð³âíîâ³ääàëåíèõ â³ä ô³êñîâàíî¿ òî÷êè F<br />
ç 0, çè 4 a ÷ ø<br />
(ôîêóñà)<br />
1<br />
³ ïðÿìî¿ y =- .<br />
4a<br />
Ðèñ. 4.24<br />
4.4.3. Ïàðàáîëà<br />
Ïàðàáîëîþ íàçèâàºòüñÿ ãåîìåòðè÷íå ì³ñöå òî÷îê ïëîùèíè,<br />
ð³âíîâ³ääàëåíèõ â³ä ô³êñîâàíî¿ òî÷êè F æ ç ,0<br />
ö çè2<br />
ø ÷<br />
(ôîêóñà) ³<br />
p<br />
p<br />
ïðÿìî¿ d (äèðåêòðèñè) x =- (ðèñ. 4.25).<br />
2<br />
Ïîçíà÷èìî ÷åðåç ð â³äñòàíü ì³æ ïðÿìîþ d ³ òî÷êîþ<br />
(p/2, 0), òîáòî ïî÷àòîê â³äë³êó ðîçòàøîâàíî â ñåðåäèí³ â³äð³çêà<br />
A⊥d. Öå îçíà÷åííÿ ïîêàçóº, ùî<br />
( ) 2 2<br />
MB = M , MB = x + p /2, M = x - p /2 + y .<br />
Òîä³ (õ + ð/2) 2 = (õ – ð/2) 2 + ó 2 àáî<br />
ó 2 = 2ðõ. (4.4.4)<br />
гâíÿííÿ (4.4.4) íàçèâàºòüñÿ êàíîí³÷íèì ð³âíÿííÿì ïàðàáîëè.<br />
Ðèñ. 4.25<br />
Âëàñòèâîñò³ ïàðàáîëè:<br />
1. Ïàðàáîëà º êðèâà äðóãîãî ïîðÿäêó.<br />
2. Îñê³ëüêè p > 0, òî ç (4.4.4) âèïëèâàº: õ ≥ 0, |y| ≥ 0.<br />
Îòæå, ïàðàáîëà º íåîáìåæåíîþ êðèâîþ, ÿêà ðîçòàøîâàíà â<br />
ïðàâ³é ï³âïëîùèí³.<br />
3. Ïàðàáîëà ìຠîäíó â³ñü ñèìåò𳿠(â³ñü Îõ).<br />
4. Ïàðàáîëà ìຠîäíó âåðøèíó, ÿêà ñï³âïàäຠç ïî÷àòêîì<br />
êîîðäèíàò.<br />
5. Åêñöåíòðèñèòåò ïàðàáîëè: ε = |M|/|MB| = 1.<br />
ÂÏÐÀÂÈ<br />
2 2 2<br />
4.20. Ïîáóäóâàòè åë³ïñ x + Ny = N . Çíàéòè éîãî ôîêóñè<br />
³ åêcöåíòðèñèòåò.<br />
2 2 2<br />
4.21. Ïîáóäóâàòè ã³ïåðáîëó x - Ny = N . Çíàéòè ¿¿ ôîêóñè,<br />
åêcöåíòðèñèòåò ³ àñèìïòîòè.<br />
2<br />
4.22. Ïîáóäóâàòè ïàðàáîëó y = 2Nx. Çíàéòè ¿¿ ôîêóñ, äèðåêòðèñó<br />
³ íàïèñàòè ð³âíÿííÿ äèðåêòðèñè.<br />
Òóò ïàðàìåòð N îçíà÷ຠ÷èñëî äàòè íàðîäæåííÿ ÷èòà÷à.<br />
126 127
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