Su'aalo Tusaale ah oo Ka Hadlaya Qaybaha Koonka Elliptical
Pendahuluan
Xisaabtu waa saynis aasaasi ah oo door muhiim ah ka ciyaara dhinacyo kala duwan oo nolosha aadanaha ah. Mawduuc gaar ah oo ku adag xisaabta waa joomatari, gaar ahaan qaybaha koonka. Maqaalkan, waxaan ka hadli doonnaa qayb ka mid ah koonka: ellipse. Maqaalkani wuxuu bixin doonaa tusaalooyin dhibaatooyin ah iyo dood dhammaystiran oo ku saabsan ellipses, kaas oo aan rajeyneyno inuu ka caawin doono ardayda inay si qoto dheer u fahmaan mowduucan.
Qeexidda iyo Sifooyinka Ellipses-ka
Kahor inta aynaan guda gelin su'aalaha tusaalaha ah, waxaa waxtar leh in marka hore la fahmo waxa uu yahay ellipse. ellipse waa ururinta dhammaan dhibcaha ku jira diyaarad oo wadarta masaafada u dhaxaysa laba dhibcood oo go'an (foci-geeda) ay joogto tahay. Labadan dhibcood ee go'an waxaa loo yaqaan foci-ga ellipse-ka (F1 iyo F2).
Qaabka aljabrada, ellipse waxaa lagu sharxi karaa isla'egtiisa guud:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
halkaas oo \( a \) ay tahay masaafada u dhaxaysa bartamaha ellipse-ka ilaa barta ugu fog ee dhidibka weyn, iyo \( b \) ay tahay masaafada u dhaxaysa bartamaha ellipse-ka ilaa barta ugu fog ee dhidibka caawiyaha.
Su'aalo Tusaale ah iyo Doodo ku saabsan Ellipses
Su'aal 1aad:
Isle'egta ellipse-ka waa \(\frac{x^2}{25} + \frac{y^2}{9} = 1\). Go'aami dhererka dhidibka weyn, dhererka dhidibka kaalmeeyaha, iyo isku-duwayaasha foci-ga.
Dood:
Isle'egta ellipse-ka la bixiyay waa \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).
1. Go'aami dhererka dhidibka ugu weyn iyo dhidibka caawiyaha:
\[ a^2 = 25 \Ferraarta Midig a = \sqrt{25} = 5 \]
\[ b^2 = 9 \Ferraarta Midig b = \sqrt{9} = 3 \]
Markaa, dhererka dhidibka weyn \(= 2a = 2(5) = 10\).
Dhererka dhidibka kaalmeeyaha \(= 2b = 2(3) = 6\).
2. Go'aami isku-duwayaasha diiradda la saarayo:
Diiradda ellipse-ku waxay ku taal dhidibka weyn meel fog oo ka fog bartamaha \(\sqrt{a^2 – b^2}\).
\[ c = \sqrt{a^2 – b^2} = \sqrt{25 – 9} = \sqrt{16} = 4 \]
Maadaama dhidibka ugu weyn ee ellipse-kan uu yahay dhidibka x, isku-duwayaasha diiradda la saarayo waa:
\( (c, 0) \) iyo \( (-c, 0) \) ama \( (4, 0) \) iyo \( (-4, 0) \).
Su'aal 2aad:
Marka la eego ellipse oo leh xarun ku taal \( (0, 0) \) iyo dhidibka weyn ee ku yaal dhidibka x, wuxuu leeyahay dherer dhidib weyn oo ah 12 iyo dherer dhidibka kaalmeeyaha ah oo ah 8. Go'aami isle'egta ellipse-ka.
Dood:
1. Marka la eego dhererka dhidibka ugu weyn \( 2a = 12 \), markaa:
\[ a = \frac{12}{2} = 6 \]
2. Marka la eego dhererka dhidibka caawinta \( 2b = 8 \), markaa:
\[ b = \frac{8}{2} = 4 \]
Isle'egta ellipse oo leh xarun ku taal \( (0, 0) \) iyo dhidibka weyn ee ku yaal dhidibka x waa:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Ku beddel \( a \) iyo \( b \) isle'egta:
\[ \frac{x^2}{6^2} + \frac{y^2}{4^2} = 1 \]
Markaa, nooca ellipse-ka waa sidan soo socota:
\[ \frac{x^2}{36} + \frac{y^2}{16} = 1 \]
Su'aal 3aad:
Go'aami kala soocidda ellipse-ka \(\frac{x^2}{49} + \frac{y^2}{36} = 1\).
Dood:
Kala soocidda (\( e \)) ee ellipse waxaa lagu bixiyaa isla'egta:
\[ e = \frac{c}{a} \]
halkaas oo \( c = \sqrt{a^2 – b^2} \).
Laga soo bilaabo isla'egta ellipse, waxaan helnaa:
\[ a^2 = 49 \Fanta midig a = 7 \]
\[ b^2 = 36 \Ferraarta Midig b = 6 \]
Hadda, waxaan helnaa \(c \):
\[ c = \sqrt{a^2 – b^2} = \sqrt{49 – 36} = \sqrt{13} \]
Kala duwanaansho (\( e \)):
\[ e = \frac{c}{a} = \frac{\sqrt{13}}{7} \]
Sidaa darteed, kala duwanaanshaha ellipse-ka waa sidan soo socota:
\[ e = \frac{\sqrt{13}}{7} \]
Su'aal 4aad:
Haddii labada qodob ee diiradda saaraya ellipse ay ku yaalliin \( (-5, 0) \) iyo \( (5, 0) \), dhererka dhidibka weyn ee ellipse-kana uu yahay 12, go'aami isle'egta ellipse-ka.
Dood:
1. Go'aami \( a \):
dhidibka weyn ee Panmaßn g waa 12, ka dibna \( 2a = 12 \).
Markaa \( a = \frac{12}{2} = 6 \).
2. Go'aami \( c \):
Labada qodob ee diiradda la saarayo waa \( (-5, 0) \) iyo \( (5, 0) \), ka dibna:
\[ c = 5 \]
3. Go'aami \( b \):
Isticmaal xiriirka \( c = \sqrt{a^2 – b^2} \):
\[ 5 = \sqrt{6^2 – b^2} \]
\[ 25 = 36 – b^2 \]
\[ b^2 = 36 – 25 \]
\[ b^2 = 11 \]
4. Soo celi isla'egta ellipse-ka:
Isle'egta ellipse waa:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Beddelka \( a \) iyo \( b \):
\[ \frac{x^2}{6^2} + \frac{y^2}{\sqrt{11}^2} = 1 \]
\[ \frac{x^2}{36} + \frac{y^2}{11} = 1 \]
Markaa, nooca ellipse-ka waa sidan soo socota:
\[ \frac{x^2}{36} + \frac{y^2}{11} = 1 \]
Xiritaanka
Iyada oo loo marayo doodda dhibaatooyinka kor ku xusan, waxaan arki karnaa in fahamka ellipses ay ku lug leedahay wax ka badan oo keliya barashada isle'egyada iyo garaafyada, laakiin sidoo kale sida sifooyinka iyo walxaha ellipses ay isku xiran yihiin. Barashada agabkan shaki la'aan waxay aad waxtar ugu yeelan doontaa dhinacyo kala duwan oo codsi ah, sida fiisigiska, xiddigiska, iyo qaybaha kale ee injineernimada. Waxaan rajeyneynaa, iyada oo loo marayo dhibaatooyinka iyo doodahan tusaalaha ah, inaad si fiican u fahmi karto fikradaha aasaasiga ah iyo codsiyada qaybaha elliptical conic.
Maqaalkan waxaa la qoray iyadoo la rajaynayo in la helo faham qoto dheer oo ku saabsan ellipses-ka. Sii wad ku celcelinta oo ha ka labalabeyn inaad sahamiso dhibaatooyin badan oo la xiriira si aad u horumariso xirfadahaaga iyo aqoontaada!
