Mehlala ea Lipotso tse Buisanang ka Likarolo tsa Elliptical Conic
Pendahuluan
Lipalo ke saense ea motheo e bapalang karolo ea bohlokoa likarolong tse fapaneng tsa bophelo ba motho. Sehlooho se seng se thata haholo lipalo ke jiometri, haholo-holo likarolo tsa conic. Sehloohong sena, re tla tšohla karolo e 'ngoe e joalo ea conic: ellipse. Sehlooho sena se tla fana ka mehlala ea mathata le puisano e felletseng ea li-ellipses, tseo re tšepang hore li tla thusa baithuti ho utloisisa sehlooho sena ka botebo.
Tlhaloso le Matlotlo a Li-ellipses
Pele re kena lipotsong tsa mohlala, ho molemo ho qala ka ho utloisisa hore na ellipse ke eng. Ellipse ke pokello ea lintlha tsohle tse sefofaneng seo kakaretso ea bolelele ba sona ho tloha lintlheng tse peli tse tsitsitseng (foci ea sona) e sa fetoheng. Lintlha tsena tse peli tse tsitsitseng li bitsoa foci ea ellipse (F1 le F2).
Ka sebopeho sa algebra, ellipse e ka hlalosoa ka equation ea eona e akaretsang:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
moo \( a \) e leng sebaka se tlohang bohareng ba ellipse ho ya ntlheng e hole ka hara axis e kgolo, mme \( b \) e leng sebaka se tlohang bohareng ba ellipse ho ya ntlheng e hole ka hara axis e thusang.
Lipotso tsa Mehlala le Puisano ea Li-ellipses
Potso ea 1:
Tekanyo ea ellipse ke \(\frac{x^2}{25} + \frac{y^2}{9} = 1\). Fumana bolelele ba axis e kholo, bolelele ba axis e thusang, le likhokahano tsa foci.
Puisano:
Tekanyo ea ellipse e fanoeng ke \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).
1. Fumana bolelele ba mothapo o ka sehloohong le mothapo o thusang:
\[ a^2 = 25 \Motsu o letona a = \sqrt{25} = 5 \]
\[ b^2 = 9 \Motsu o Motshehare b = \sqrt{9} = 3 \]
Kahoo, bolelele ba axis e kholo \(= 2a = 2(5) = 10\).
Bolelele ba mothapo o thusang \(= 2b = 2(3) = 6\).
2. Fumana likhokahano tsa ho tsepamisa maikutlo:
Sepheo sa ellipse se hodima axis e kgolo e hole le setsi sa \(\sqrt{a^2 – b^2}\).
\[ c = \sqrt{a^2 – b^2} = \sqrt{25 – 9} = \sqrt{16} = 4 \]
Kaha motsoako o moholo oa ellipse ena ke motsoako oa x, likhokahano tsa tsepamiso ke:
\( (c, 0) \) le \( (-c, 0) \) kapa \( (4, 0) \) le \( (-4, 0) \).
Potso ea 2:
Ha ho fanoe ka ellipse e nang le setsi ho \( (0, 0) \) le axis e kholo ho x-axis, e na le bolelele ba axis e kholo ea 12 le bolelele ba axis e thusang ea 8. Fumana equation ea ellipse.
Puisano:
1. Ha ho nahanoa ka bolelele ba mothapo o ka sehloohong \( 2a = 12 \), joale:
\[ a = \frac{12}{2} = 6 \]
2. Ha ho nahanoa ka bolelele ba mothapo o thusang \( 2b = 8 \), joale:
\[ b = \frac{8}{2} = 4 \]
Tekanyo ea ellipse e nang le setsi ho \( (0, 0) \) le axis e kholo ho x-axis ke:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Kenya \( a \) le \( b \) sebakeng sa equation:
\[ \frac{x^2}{6^2} + \frac{y^2}{4^2} = 1 \]
Kahoo, equation ea ellipse ke:
\[ \frac{x^2}{36} + \frac{y^2}{16} = 1 \]
Potso ea 3:
Fumana hore na ellipse e fapane hakae \(\frac{x^2}{49} + \frac{y^2}{36} = 1\).
Puisano:
Ho se tshwane (\( e \)) ha ellipse ho fanwa ke equation:
\[ e = \frac{c}{a} \]
moo \( c = \sqrt{a^2 – b^2} \).
Ho tsoa ho equation ea ellipse, re fumana:
\[ a^2 = 49 \Motsu o ka letsohong le letona a = 7 \]
\[ b^2 = 36 \Motsu o ka letsohong le letona b = 6 \]
Jwale, re fumana \( c \):
\[ c = \sqrt{a^2 – b^2} = \sqrt{49 – 36} = \sqrt{13} \]
Ho ikamahanya le maemo (\( e \)):
\[ e = \frac{c}{a} = \frac{\sqrt{13}}{7} \]
Kahoo, ho se hlaka ha ellipse ke:
\[ e = \frac{\sqrt{13}}{7} \]
Potso ea 4:
Haeba lintlha tse peli tsa focus tsa ellipse li fumaneha ho \( (-5, 0) \) le \( (5, 0) \), 'me bolelele ba axis e kholo ea ellipse ke 12, fumana equation ea ellipse.
Puisano:
1. Fumana \( a \) :
Panmaßn g major axis ke 12, ebe \( 2a = 12 \).
Kahoo \( a = \frac{12}{2} = 6 \).
2. Fumana \( c \) :
Lintlha tse peli tsa bohlokoa ke \( (-5, 0) \) le \( (5, 0) \), ebe:
\[c = 5 \]
3. Fumana \( b \) :
Sebelisa kamano \( c = \sqrt{a^2 – b^2} \):
\[5 = \sqrt{6^2 – b^2} \]
\[25 = 36 – b^2 \]
\[ b^2 = 36 – 25 \]
\[ b^2 = 11 \]
4. Khutlisa equation ea ellipse:
Tekanyo ea ellipse ke:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Ho nkela \( a \) le \( b \ sebaka):
\[ \frac{x^2}{6^2} + \frac{y^2}{\sqrt{11}^2} = 1 \]
\[ \frac{x^2}{36} + \frac{y^2}{11} = 1 \]
Kahoo, equation ea ellipse ke:
\[ \frac{x^2}{36} + \frac{y^2}{11} = 1 \]
Ho koala
Ka puisano ea mathata a kaholimo, re ka bona hore ho utloisisa li-ellipse ho kenyelletsa ho fetang feela ho ithuta li-equation le li-graph tsa tsona, empa hape le kamoo thepa le likarolo tsa li-ellipse li amanang kateng. Ho tseba boitsebiso bona ntle ho pelaelo ho tla ba molemo haholo masimong a fapaneng a ts'ebeliso, joalo ka fisiks, bolepi ba linaleli le masimo a mang a boenjiniere. Re tšepa hore, ka mathata le lipuisano tsena tsa mehlala, u ka utloisisa hamolemo likhopolo tsa motheo le ts'ebeliso ea likarolo tsa conic tsa elliptical.
Sengoloa sena se ngotsoe ka tšepo ea ho fana ka kutloisiso e tebileng ea li-ellipses. Tsoela pele ho ikoetlisa 'me u se ke ua tsilatsila ho hlahloba mathata a mang a amanang le ona ho ntlafatsa tsebo le tsebo ea hau!
