Su'aalo Tusaale ah oo Ka Hadlaya Qaybaha Kooniga ee Hyperbolic
Pendahuluan
Xisaabta, qaybta konik, oo inta badan loogu yeero qaybta konik, waa qalooc laga helo isgoyska koonka iyo diyaaradda. Waxaa jira afar nooc oo waaweyn oo qaybaha konik ah: goobo, ellipses, parabolas, iyo hyperbolas. Maqaalkan, waxaan diiradda saari doonnaa hyperbola, nooc ka mid ah qaybta konik ah oo leh codsiyo badan oo ku saabsan qaybaha sida xiddigiska, fiisigiska, iyo injineernimada. Maqaalkani wuxuu soo bandhigi doonaa tusaalooyin dhibaatooyin ah iyo dooddooda ku saabsan mowduucan, iyadoo ujeedadu tahay in la caawiyo akhristayaasha inay fahmaan fikradda iyo sida loo xalliyo dhibaatooyinka la xiriira hyperbolas.
Qeexidda iyo Sifooyinka Hyperbole
Kahor inta aynaan guda gelin su'aalaha tusaalaha ah, aan marka hore ka wada hadalno qaar ka mid ah fikradaha aasaasiga ah ee ku saabsan hyperbola.
Hyperbola waa goobta dhibcaha ee diyaarad si farqiga masaafada ee dhibic kasta iyo laba dhibcood oo go'an (oo loo yaqaan foci) uu noqdo mid joogto ah.
Isle'egta guud ee hyperbola ee qaabka caadiga ah waa:
\[ \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1 \]
ama
\[ \frac{y^2}{b^2} – \frac{x^2}{a^2} = 1 \]
Halkee:
– \(a\) waa masaafada u dhaxaysa bartamaha hyperbola ilaa meesha ugu sarreysa (geeska).
– \(b\) waa masaafada la xiriirta masaafada laga bilaabo bartamaha ilaa barta ugu dhow ee asymptote-ka hyperbola.
Hyperbola-ga oo si toosan isu gudba, qaabka guud ee la isticmaalay waa:
\[ \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1 \]
Dhanka kale, hyperbolas-ka si toosan isu gudba:
\[ \frac{y^2}{b^2} – \frac{x^2}{a^2} = 1 \]
Su'aalo iyo Doodo Tusaale ah
Su'aal 1aad:
Marka la eego isleegta hyperbola \( \frac{x^2}{16} – \frac{y^2}{9} = 1 \). Go'aami:
1. Xarunta hyperbola.
2. Dhererka dhidibka ugu weyn iyo dhidibka labaad.
3. Barta diiradda la saarayo.
4. Isle'egta Asymptote.
5. Sawir hyperbola-da.
Dood:
1. Xarunta Hyperbola:
Maadaama qaabka isle'egta kor ku xusan uu yahay mid caadi ah oo aysan jirin ereyo \((x - h)\) ama \((y - k)\), bartamaha hyperbola-gan wuxuu ku yaal barta (0,0).
2. Dhererka Dhidibka Weyn iyo Dhidibka Labaad:
Laga soo bilaabo isla'egta \( \frac{x^2}{16} – \frac{y^2}{9} = 1 \), waxaa la ogyahay in:
\[
a^2 = 16 \Fanta midig a = 4
\]
\[
b^2 = 9 \Ferraarta Midig b = 3
\]
Dhererka dhidibka weyn waa \(2a = 2 \jeer 4 = 8\).
Dhererka dhidibka labaad waa \(2b = 2 \jeer 3 = 6 \).
3. Barta Diiradda:
Si aan u helno barta diiradda la saarayo, waxaan isticmaalnaa xiriirka:
\[
c^2 = a^2 + b^2
\]
\[
c^2 = 16 + 9 = 25 \Ferraarta Midig c = \sqrt{25} = 5
\]
Maadaama buundadani ay tahay mid jiif ah, meelaha ugu muhiimsan waxay ku yaalliin dhibcaha \((\pm c, 0)\), kuwaas oo kala ah \((5, 0)\) iyo \(-5, 0)\).
4. Isle'egta Asymptote:
Asymptote waa xariiq toosan oo u dhow hyperbola. Isla'egtan caadiga ah, asymptote-ka waxaa lagu go'aamin karaa:
\[
y = \pm \frac{b}{a}x \Midigta usha y = \pm \frac{3}{4}x
\]
Markaa, isle'egyada asymptote waa \( y = \frac{3}{4}x \) iyo \( y = -\frac{3}{4}x \).
5. Sawirka Hyperbola:
Si aan u sharaxno hyperbola, waxaan u baahanahay:
– Waxay calaamadeysaa bartamaha (0,0).
– U diyaari meelaha ugu sarreeya dhibcaha (4,0) iyo (-4,0).
– Sawir calaamado aan calaamado lahayn oo leh xariiqyo y = (3/4)x iyo y = -(3/4)x oo dhex maraya bartamaha.
– Calaamadee meelaha diiradda la saarayo (5,0) iyo (-5,0).
Su'aal 2aad:
Go'aami isle'egta hyperbola oo leh dhidib weyn oo dhererkiisu yahay 10 cutub, dhidib labaad oo dhererkiisu yahay 8 cutub, oo xuddun u ah asalka.
Dood:
Su'aasha laga qabo waxaa la ogyahay in dhererka dhidibka ugu weyn (2a) uu yahay 10 cutub, ka dibna:
\[ 2a = 10 \Qalab toos ah a = 5 \]
Dhererka dhidibka labaad (2b) waa 8 cutub, sidaas darteed:
\[ 2b = 8 \Ferraarta Midig b = 4 \]
Iyadoo bartamaha laga helayo asalka (0,0), waxaan u qori karnaa isla'egta caadiga ah ee hyperbola sida soo socota:
\[ \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1 \]
Ka dib markaad beddesho qiimayaasha a iyo b:
\[ \frac{x^2}{25} – \frac{y^2}{16} = 1 \]
Markaa, isle'egta hyperbola ee su'aasha laga qabo waa:
\[ \frac{x^2}{25} – \frac{y^2}{16} = 1 \]
Su'aal 3aad:
Marka la eego hyperbola toosan oo leh isle'egta \(\frac{y^2}{36} – \frac{x^2}{16} = 1 \). Go'aami masaafada u dhaxaysa labadeeda qodob.
Dood:
Isleegta hyperbola \(\frac{y^2}{36} – \frac{x^2}{16} = 1\), waxaan aqoonsaneynaa taas:
\[ a^2 = 36 \Fanta midig a = 6 \]
\[ b^2 = 16 \Ferraarta Midig b = 4 \]
Si loo helo masaafada u dhaxaysa laba qodob, waxaan isticmaalnaa xiriirka:
\[ c^2 = a^2 + b^2 \]
\[ c^2 = 36 + 16 = 52 \Ferraarta Midig c = \sqrt{52} = 2\sqrt{13} \]
Masaafada u dhaxaysa labada qaybood ee hyperbola waxaa loo xisaabiyaa 2 jeer masaafada ay qaybaha dhexe ka jiraan:
\[ 2c = 2 \ jeer 2 \ sqrt{13} = 4 \ sqrt{13} \]
Markaa, masaafada u dhaxaysa labada qaybood waa halbeegyo \(4\sqrt{13}\).
Gabagabo
Maqaalkan, waxaan kaga hadalnay dhowr tusaale oo dhibaatooyin ah oo ku saabsan hyperbolas, oo ay ku jiraan aqoonsashada bartamaha, dhererka dhidibka waaweyn iyo kuwa yaryar, diiradda, isle'egyada asymptotes-ka, iyo garaafaynta hyperbolas-ka. Fahmidda sida loo xalliyo dhibaatooyinkan waa muhiim, gaar ahaan ardayda baranaysa joomatari falanqayn ama xisaabta horumarsan.
Hyperbola ma aha oo kaliya aragti, laakiin sidoo kale waxay leedahay codsiyo ballaaran oo ku saabsan qaybaha kale ee sayniska sida xiddigiska, radar, iyo GPS. Sidaa darteed, barashada hyperbola ma aha oo kaliya xalinta dhibaatooyinka xisaabta, laakiin sidoo kale fahamka sida fikradahan xisaabta loogu dabaqi karo nolosha dhabta ah.
