Equation yeEllipse muGeometry
Dura remaonde (ellipse) igwara rakakosha mu geometry, rinoonekwa munzvimbo dzakasiyana-siyana, kubva pamasvomhu chaiwo kusvika kumashandisirwo mufizikisi, mainjiniya, uye nyeredzi. Zvichitaurwa zviri nyore, dura remaonde rinogona kunzwisiswa se "denderedzwa rakatambanudzwa" kuitira kuti rive rakareba munzira imwe chete. Zvisinei, tsananguro yepamutemo ye dura remaonde inonyanya kufadza: dura remaonde ndiro seti yemapoinzi ese ari mundege umo huwandu hwedaro rawo kubva kumapoinzi maviri akagadzika (anonzi foci) hunogara huchienderana. Kubva pane iyi tsananguro, equation ye dura remaonde inogona kuwanikwa uye kudzidzwa, mumafomu akajairwa uye akajairika.
1. Kunzwisisa maEllipses neZvinhu Zvawo
Kuti tinzwisise equation ye ellipse, tinofanira kuziva zvinhu zvikuru zve ellipse:
1. Pakati pedenderedzwa (pakati): pakati pedenderedzwa, rinowanzo fananidzirwa \((h, k)\).
2. Chidimbu chikuru: dhayamita refu kwazvo yedenderedzwa.
3. Minor axis: dhayamita pfupi ye ellipse yakatarisana ne major axis.
4. Kutarisisa (foci): mapoinzi maviri akasimba anoshanda sereferensi yekutsanangurwa kwe ellipse, inowanzo ratidzwa \(F_1\) uye \(F_2\).
5. Semimajor radius: hafu yehurefu hwe major axis, inomiririrwa \(a\).
6. Seminor radius: hafu yehurefu hwe minor axis, inonongedzwa \(b\).
7. Daro kubva pakati kusvika pakutarisa: rinoratidza \(c\), rine hukama hwe elliptical:
\[
c^2 = a^2 – b^2
\]
Kusawirirana kwepfungwa kunowanzoitika pano: mudenderedzwa, \(a \ge b\) inogara ichibata uye chinangwa chiri padivi guru.
Pamusoro pezvo, kune pfungwa yekusafanana \(e\) iyo inoyera "kutsveyama kwekunze" kwe 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) Akisi huru yakatwasuka
Kana axis huru iri parallel ney-axis, saka:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
ne \(a > b\). Chinangwa chiri pa \(y\)-axis, kureva:
\[
(0, \pm c), \quad c^2 = a^2 – b^2
\]
Fomu iri rinoita kuti zvive nyore kuverenga hunhu hwe ellipse: kukosha kwe \(a\) uye \(b\) kunoratidza zvakananga saizi ye ellipse, ukuwo \(c\) ichisarudza nzvimbo ye foci.
3. Equation yeEllipse Yakatarisana pa \((h,k)\)
Mumatambudziko mazhinji ekuongorora geometry, ellipse haisi nguva dzose iri pakati pe coordinate centre. Kana ellipse iri pakati pa \((h,k)\), saka standard equation inoshanduka kuita:
a) Akisi huru yakatambanuka
\[
\frac{(xh)^2}{a^2} + \frac{(yk)^2}{b^2} = 1
\]
b) Akisi huru yakatwasuka
\[
\frac{(xh)^2}{b^2} + \frac{(yk)^2}{a^2} = 1
\]
Kuchinja uku kunongova kushandurwa kwe ellipse, iyo pakutanga yaive yakatarisana nekwaitangira. Chinangwa chinotamirawo kunzvimbo itsva:
– Kune horizontal major axis: \((h \pm c, k)\)
– Kune axis huru yakamira: \((h, k \pm c)\)
4. Kubva paTsananguro yeKutarisa Kusvika paEquation yeEllipse
Tsanangudzo ye ellipse sehuwandu hwemadaro kusvika ku foci mbiri dzisingachinji inogona kushandiswa sehwaro hwekuwana equation. Semuenzaniso, ngatitii foci dziri pa \((c,0)\) uye \((-c,0)\), uye poindi iri pa ellipse iri \((x,y)\). Madaro epoindi iyoyo kune imwe neimwe focus ndeaya:
\[
d_1 = \sqrt{(xc)^2 + y^2}, \quad d_2 = \sqrt{(x+c)^2 + y^2}
\]
Sezvo huwandu hwacho huchigara huripo:
\[
d_1 + d_2 = 2a
\]
Nekushandisa algebraic manipulation (kuumba squared kaviri kubvisa midzi), tinowana equation:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
ne \(b^2 = a^2 – c^2\). Izvi zvinoratidza kuti chimiro che ellipse hachisi che "kurangarira" chete, asi chinobva mutsananguro yejometri.
5. General Equation yeEllipse uye Kuzivikanwa Kwayo
Mukuita, tinowanzo sangana nema quadratic equations ane ma variable maviri asiri muchimiro chakajairika, semuenzaniso:
\[
Ax^2 + By^2 + Cx + Dy + E = 0
\]
Equation yakaita seiyi inogona kumiririra ellipse, parabola, kana hyperbola. Kuti ive nechokwadi chekuti iellipse (ine axes dzakafanana ne coordinates), kazhinji \(A\) uye \(B\) zvinofanira kunge zviri:
- chiratidzo chimwe chete (chakanaka kana kuti chisina kunaka),
– uye kazhinji kwete saizi imwe chete (kana dziri saizi imwe chete uye pasina izwi \(xy\), zvingangoita kuti chimiro chacho chive denderedzwa).
Kuti uchinje kuita fomu re ellipse rakajairika, nzira inonyanya kushandiswa ndeyekupedzisa sikweya pamashoko ekuti \(x\) uye \(y\). Muenzaniso uri nyore:
\[
4x^2 + 9y^2 – 8x + 18y – 5 = 0
\]
Boka:
\[
4(x^2 – 2x) + 9(y^2 + 2y) = 5
\]
Pedzisa chikwere:
\[
4[(x-1)^2 – 1] + 9[(y+1)^2 – 1] = 5
\]
\[
4(x-1)^2 + 9(y+1)^2 = 5 + 4 + 9 = 18
\]
Kwemakore gumi nemasere:
\[
\frac{(x-1)^2}{\frac{18}{4}} + \frac{(y+1)^2}{2} = 1
\]
inova ndiyo fomu yakajairika ye ellipse ine pakati \((1,-1)\).
6. Mashandisirwo eEllipses muGeometry neHupenyu Hwechokwadi
Ma "ellipses" haasi zvinhu zvepfungwa chete. Mu geometry nesainzi inoshandiswa, ma "ellipses" anoita basa guru:
1. Nyanzvi dzenyeredzi (Mutemo waKepler): kutenderera kwenyika kwakaita sedenderedzwa, zuva riri panzvimbo imwe chete.
2. Optics uye acoustics: hunhu hwe elliptical reflection hunotaura kuti mafungu anobva pane imwe focus acharatidzwa kuburikidza neimwe focus. Izvi zvinoshandiswa mukugadzira ma concert holls kana mamwe magirazi e reflector.
3. Uinjiniya hwemakanika: mamwe magiya kana kuti michina yemakamera anoshandisa nzira dzakaita sedenderedzwa.
4. Magadzirirwo: chimiro chedenderedzwa chinopa musanganiswa werunako uye mashandiro ekurira.
Nekunzwisisa equation ye ellipse, tinogona kuongorora saizi, nzvimbo, uye hunhu hwematanho ekufamba muhurongwa hwakasiyana-siyana.
7. Kesimpulan
Equation ye ellipse mu geometry inovhara musiyano uripo pakati petsananguro ye geometri (huwandu hwemadaro kusvika ku foci mbiri dzisingachinji) uye chimiro che analytical (algebraic equation mu coordinates). Chimiro che standard che ellipse chinoita kuti zvive nyore kuziva pakati, hurefu hwe axes, nenzvimbo dze foci, nepo mafomu akajairika anogona kushandurwa kuita chimiro chakajairwa nekuzadza sikweya. Kunzwisisa ellipses hakungobatsiri chete kugadzirisa matambudziko e analytical geometry asiwo kunovhura ruzivo rwekuti masvomhu anotsanangura sei zviitiko zvechisikigo zvakaita se planetary orbits uye hunhu hwe wave reflection.
Kana muchida, ndinogonawo kuwedzera mienzaniso yezvinetso uye kupedzisa hurukuro (semuenzaniso kuona chinangwa, kusanzwisisika, kana kudhirowa mufananidzo we ellipse kubva mu equation yayo).
