Equation ea Ellipse ho Geometry
Ellipse ke mothapo oa bohlokoa oa jeometri, o hlahang maemong a fapaneng a fapaneng, ho tloha lipalo tse hloekileng ho ea lits'ebetsong tsa fisiks, boenjiniere le bolepi ba linaleli. Ka mantsoe a bonolo, ellipse e ka utloisisoa e le "selikalikoe se otlolohileng" e le hore se be selelele ka lehlakoreng le le leng. Leha ho le joalo, tlhaloso ea semmuso ea ellipse e khahlisa haholo: ellipse ke sete ea lintlha tsohle tse sefofaneng seo kakaretso ea libaka tsa tsona ho tloha lintlheng tse peli tse tsitsitseng (tse bitsoang foci) e lulang e sa fetohe. Ho tsoa tlhalosong ena, equation ea ellipse e ka fumanoa le ho ithutoa, ka bobeli ka mekhoa e tloaelehileng le e akaretsang.
1. Ho Utloisisa Li-ellipse le Likarolo tsa Tsona
Ho utloisisa equation ea ellipse, re hloka ho tseba likarolo tsa mantlha tsa ellipse:
1. Bohareng ba ellipse (bohareng): ntlha e bohareng ea ellipse, hangata e tšoantšetsoa \((h, k)\).
2. Mothapo o moholo: bophara bo bolelele ka ho fetisisa ba ellipse.
3. Mothapo o monyane: bophara bo bokgutshwane ba ellipse bo otlolohileng ho mothapo o moholo.
4. Tsepamiso (foci): lintlha tse peli tse tsitsitseng tse sebetsang e le litšupiso bakeng sa tlhaloso ea ellipse, hangata e bontšoang \(F_1\) le \(F_2\).
5. Semimajor radius: halofo ea bolelele ba axis e kholo, e tšoantšetsoang \(a\).
6. Seminor radius: halofo ea bolelele ba axis e nyane, e bontšitsoeng \(b\).
7. Sebaka ho tloha bohareng ho ea ho tsepamiso: se bontšitsoeng \(c\), ka kamano e tloaelehileng ea elliptical:
\[
c^2 = a^2 – b^2
\]
Hangata ho hlaha khohlano ea mehopolo mona: ka ellipse, \(a \ge b\) e lula e tšoara 'me tsepamiso e lutse mothating o moholo.
Ho phaella moo, ho na le mohopolo wa ho se tshwane \(e\) o lekanyang "tshekamelo ya kantle" ya ellipse:
\[
e = \frac{c}{a}, \quad 0 \le e < 1
\]
Jika \(e = 0\), elips menjadi lingkaran (karena \(c = 0\), fokus berimpit di pusat).
2. Persamaan Standar Elips Berpusat di Titik Asal
Jika elips berpusat di titik asal \((0,0)\) dan sumbu-sumbunya sejajar sumbu koordinat, persamaan elips memiliki bentuk standar yang sangat dikenal.
a) Sumbu mayor horizontal
Jika sumbu mayor sejajar sumbu-\(x\), maka:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
dengan \(a > b\). Fokus terletak pada sumbu-\(x\), yaitu di titik:
\[
(\pm c, 0), \quad \text{with} c^2 = a^2 – b^2
\]
b) Mokhahlelo o moholo o otlolohileng
Haeba axis e kholo e bapile le axis ea y, joale:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
ka \(a > b\). Sepheo se ho \(y\)-axis, e leng:
\[
(0, \pm c), \quad c^2 = a^2 – b^2
\]
Sebopeho sena se tloaelehileng se etsa hore ho be bonolo ho bala litšobotsi tsa ellipse: boleng ba \(a\) le \(b\) bo bontša ka ho toba boholo ba ellipse, ha \(c\) e khetholla boemo ba foci.
3. Equation ea Ellipse e Bohareng ho \((h,k)\)
Mathata a mangata a jeometri ea tlhahlobo, ellipse ha se kamehla e leng bohareng ba setsi sa likhokahano. Haeba ellipse e bohareng ba \((h,k)\), joale equation e tloaelehileng e fetoha ho:
a) Mokhahlelo o moholo o otlolohileng
\[
\frac{(xh)^2}{a^2} + \frac{(yk)^2}{b^2} = 1
\]
b) Mokhahlelo o moholo o otlolohileng
\[
\frac{(xh)^2}{b^2} + \frac{(yk)^2}{a^2} = 1
\]
Phetoho ena ha e le hantle ke phetoho feela (phetolelo) ea ellipse, eo qalong e neng e le bohareng ba tšimoloho. Sepheo se boetse se fetela setsing se secha:
– Bakeng sa axis e kholo e rapameng: \((h \pm c, k)\)
– Bakeng sa axis e kgolo e otlolohileng: \((h, k \pm c)\)
4. Ho tloha Tlhalosong ea Tsepamiso ho ea ho Tekanyo ea Ellipse
Tlhaloso ea ellipse e le kakaretso ea libaka tse eang ho foci tse peli tse sa fetoheng e ka sebelisoa e le motheo oa ho fumana li-equation. Mohlala, a re re li-foci li ho \((c,0)\) le \((-c,0)\), 'me ntlha ho ellipse ke \((x,y)\). Libaka tsa ntlha eo ho ea ho tsepamiso ka 'ngoe ke:
\[
d_1 = \sqrt{(xc)^2 + y^2}, \quad d_2 = \sqrt{(x+c)^2 + y^2}
\]
Kaha tjhelete eo ha e fetohe:
\[
d_1 + d_2 = 2a
\]
Ka ho laola algebra (ho etsa sekwere habeli ho tlosa metso), re fumana equation:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
ka \(b^2 = a^2 – c^2\). Sena se bontsha hore sebopeho se tlwaelehileng sa ellipse ha se feela foromo ya "ho hopola", empa ha e le hantle e tswa tlhalosong ya jeometri.
5. Kakaretso ea Tekanyo ea Ellipse le Boitsebiso ba Eona
Ha re ntse re sebetsa, hangata re kopana le li-quadratic equation tse nang le li-variable tse peli tse seng ka mokhoa o tloaelehileng, mohlala:
\[
Ax^2 + Ka^2 + Cx + Dy + E = 0
\]
Tekanyo e kang ena e ka emela ellipse, parabola, kapa hyperbola. Ho netefatsa hore ke ellipse (e nang le li-axes tse bapileng le li-coordinates), hangata \(A\) le \(B\) e tlameha ho ba:
– letshwao le tshwanang (le letle kapa le lebe),
– mme ka kakaretso ha di tshwane ka boholo (haeba di lekana ka boholo mme ho se na lentswe \(xy\), ho ka etsahala hore sebopeho ke sedikadikwe).
Ho fetolela ho foromo e tloaelehileng ea ellipse, mokhoa o sebelisoang haholo ke ho tlatsa sekwere ka mantsoe a \(x\) le \(y\). Mohlala o bonolo:
\[
4x^2 + 9y^2 – 8x + 18y – 5 = 0
\]
Sehlopha:
\[
4(x^2 – 2x) + 9(y^2 + 2y) = 5
\]
Qetella sekwere:
\[
4[(x-1)^2 – 1] + 9[(y+1)^2 – 1] = 5
\]
\[
4(x-1)^2 + 9(y+1)^2 = 5 + 4 + 9 = 18
\]
Bakeng sa 18:
\[
\frac{(x-1)^2}{\frac{18}{4}} + \frac{(y+1)^2}{2} = 1
\]
e leng sebopeho se tloaelehileng sa ellipse e nang le setsi \((1,-1)\).
6. Ts'ebeliso ea Li-ellipses ho Jeometri le Bophelo ba Sebele
Li-ellipse ha se lintho tsa khopolo-taba feela. Ho jeometri le saense e sebelisitsoeng, li-ellipse li bapala karolo e kholo:
1. Bolepi ba linaleli (Molao oa Kepler): potoloho ea polanete e elliptical ka Letsatsi le shebane le le leng.
2. Optics le acoustics: thepa ea ho bonahatsa ka elliptical e bolela hore maqhubu a tsoang ho tsepamiso e 'ngoe a tla bonahatsoa ka tsepamiso e 'ngoe. Sena se sebelisoa ho rala liholo tsa konsarete kapa liipone tse itseng tsa ho bonahatsa.
3. Boenjiniere ba mechini: mechine e itseng ea lisebelisoa kapa ea cam e sebelisa litsela tse elliptical.
4. Meralo: sebopeho sa elliptical se fana ka motsoako oa botle le mosebetsi oa molumo.
Ka ho utloisisa equation ea ellipse, re ka sekaseka boholo, boemo le thepa ea litsela litsamaisong tse fapaneng.
7. Kesimpulan
Tekanyo ea ellipse ho geometry e kopanya sekheo pakeng tsa tlhaloso ea jeometri (kakaretso ea libaka tse peli tse sa fetoheng) le boemeli ba tlhahlobo (equation ea algebraic ka har'a li-coordinate). Sebopeho se tloaelehileng sa ellipse se etsa hore ho be bonolo ho khetholla setsi, bolelele ba li-axes, le maemo a foci, ha libopeho tse akaretsang li ka fetoloa ho ba sebopeho se tloaelehileng ka ho tlatsa sekwere. Ho utloisisa li-ellipse ha ho thuse feela ho rarolla mathata a jeometri ea tlhahlobo empa hape ho bula temohisiso ea hore na lipalo li hlalosa liketsahalo tsa tlhaho joang joalo ka lipotoloho tsa lipolanete le thepa ea ho bonahatsa maqhubu.
Haeba o batla, nka eketsa mehlala ea mathata le ho phethela lipuisano (mohlala, ho fumana sepheo, ho se tšoane, kapa ho taka setšoantšo sa ellipse ho tsoa ho equation ea eona).
