Mibvunzo Yemuenzaniso Yekukurukura Zvikamu zveElliptical Conic
Pendauluan
Masvomhu isainzi inokosha ine basa rakakosha muhupenyu hwevanhu. Imwe nyaya inonyanya kunetsa mumasvomhu igeometry, kunyanya zvikamu zveconic. Muchinyorwa chino, tichakurukura chimwe chikamu chakadaro checonic: ellipse. Chinyorwa chino chichapa mienzaniso yezvinetso uye hurukuro yakazara yeellipses, izvo zvatinotarisira kuti zvichabatsira vadzidzi kunzwisisa nyaya iyi zvakadzama.
Tsanangudzo uye Hunhu hweEllipses
Tisati tapinda mumibvunzo yemuenzaniso, zvakanaka kutanga tanzwisisa kuti chii chinonzi ellipse. Ellipse muunganidzwa wemapoinzi ese ari mundege ane huwandu hwedaro kubva kumapoinzi maviri akagadzika (foci yayo) hunogara huripo. Mapoinzi maviri aya akagadzika anonzi foci yeellipse (F1 naF2).
Muchimiro chealgebraic, ellipse inogona kutsanangurwa ne equation yayo yakajairika:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
apo \( a \) iri daro kubva pakati pe ellipse kusvika panzvimbo iri kure pane axis huru, uye \( b \) iri daro kubva pakati pe ellipse kusvika panzvimbo iri kure pane axis yekubatsira.
Mibvunzo yeMienzaniso neKukurukurirana kweEllipses
Mubvunzo 1:
Equation ye ellipse ndeye \(\frac{x^2}{25} + \frac{y^2}{9} = 1\). Sarudza kureba kwe major axis, kureba kwe assistant axis, uye ma coordinates e foci.
Kukurukurirana:
Equation ye ellipse yakapihwa ndeye \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).
1. Sarudza kureba kwe axis huru ne axis yekubatsira:
\[ a^2 = 25 \Museve wekurudyi a = \sqrt{25} = 5 \]
\[ b^2 = 9 \Museve wekurudyi b = \sqrt{9} = 3 \]
Saka, kureba kwe major axis \(= 2a = 2(5) = 10\).
Kureba kweakisi yekubatsira \(= 2b = 2(3) = 6\).
2. Sarudza macoordinates ekutarisa:
Chinhu chinonyanya kutariswa nedenderedzwa remhino chiri pa axis huru iri kure nepakati pe \(\sqrt{a^2 – b^2}\).
\[ c = \sqrt{a^2 – b^2} = \sqrt{25 – 9} = \sqrt{16} = 4 \]
Sezvo axis huru yedenderedzwa iri iri x-axis, macoordinates efocus ndeaya:
\( (c, 0) \) uye \( (-c, 0) \) kana \( (4, 0) \) uye \( (-4, 0) \).
Mubvunzo 2:
Kana ukashandisa ellipse ine pakati pa \( (0, 0) \) uye major axis pa x-axis, ine major axis length ye12 uye assistant axis length ye8. Sarudza equation yeellipse.
Kukurukurirana:
1. Zvichienderana nehurefu hwe axis huru \( 2a = 12 \), zvino:
\[ a = \frac{12}{2} = 6 \]
2. Zvichienderana nehurefu hwe assistant axis \( 2b = 8 \), zvino:
\[ b = \frac{8}{2} = 4 \]
Equation ye ellipse ine pakati pa \( (0, 0) \) uye major axis pa x-axis ndeiyi:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Isa \( a \) uye \( b \) muequation:
\[ \frac{x^2}{6^2} + \frac{y^2}{4^2} = 1 \]
Saka, equation ye ellipse ndeiyi:
\[ \frac{x^2}{36} + \frac{y^2}{16} = 1 \]
Mubvunzo 3:
Tsvaga kuti ellipse haina kufanana ne ellipse \(\frac{x^2}{49} + \frac{y^2}{36} = 1\).
Kukurukurirana:
Kusafanana (\( e \)) kwe ellipse kunopiwa ne equation:
\[ e = \frac{c}{a} \]
apo \( c = \sqrt{a^2 – b^2} \).
Kubva mu equation ye ellipse, tinowana:
\[ a^2 = 49 \Museve wekurudyi a = 7 \]
\[ b^2 = 36 \Museve wekurudyi b = 6 \]
Zvino, tinowana \( c \):
\[ c = \sqrt{a^2 – b^2} = \sqrt{49 – 36} = \sqrt{13} \]
Kusawirirana (\( e \)):
\[ e = \frac{c}{a} = \frac{\sqrt{13}}{7} \]
Saka, kusiyana kwe ellipse ndekwekuti:
\[ e = \frac{\sqrt{13}}{7} \]
Mubvunzo 4:
Kana nzvimbo mbiri dze ellipse dziri pa \( (-5, 0) \) uye \( (5, 0) \), uye kureba kwe major axis ye ellipse kuri 12, sarudza equation ye ellipse.
Kukurukurirana:
1. Sarudza \( a \) :
Panmaßn g major axis i12, wobva \( 2a = 12 \).
Saka \( a = \frac{12}{2} = 6 \).
2. Sarudza \( c \) :
Pfungwa mbiri dzinonyanya kutariswa ndi \( (-5, 0) \) uye \( (5, 0) \), zvino:
\[c = 5 \]
3. Sarudza \( b \) :
Shandisa hukama \( c = \sqrt{a^2 – b^2} \):
\[ 5 = \sqrt{6^2 – b^2} \]
\[ 25 = 36 – b^2 \]
\[ b^2 = 36 – 25 \]
\[ b^2 = 11 \]
4. Dzorera equation ye ellipse:
Equation ye ellipse ndeiyi:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Kutsiva \( a \) uye \( b \):
\[ \frac{x^2}{6^2} + \frac{y^2}{\sqrt{11}^2} = 1 \]
\[ \frac{x^2}{36} + \frac{y^2}{11} = 1 \]
Saka, equation ye ellipse ndeiyi:
\[ \frac{x^2}{36} + \frac{y^2}{11} = 1 \]
Penutup
Kuburikidza nekukurukura matambudziko ari pamusoro apa, tinogona kuona kuti kunzwisisa ma ellipses kunosanganisira zvinopfuura kungodzidza ma equation nema graphs avo, asiwo kuti hunhu nezvinhu zve ellipses zvine hukama sei. Kuziva chinhu ichi pasina mubvunzo kuchabatsira zvikuru muzvikamu zvakasiyana-siyana zvekushandisa, zvakaita sefizikisi, nyeredzi, nedzimwe nzvimbo dzeinjiniya. Tinovimba, kuburikidza nematambudziko aya emuenzaniso nehurukuro, unogona kunzwisisa zviri nani pfungwa huru uye mashandisirwo ezvikamu zve elliptical conic.
Chinyorwa ichi chakanyorwa netariro yekukupa kunzwisisa kwakadzama kwe ellipses. Ramba uchidzidzira uye usazeza kutsvaga mamwe matambudziko ane chekuita nazvo kuti uvandudze hunyanzvi hwako neruzivo rwako!
