Mibvunzo Yemuenzaniso Inokurukura Zvikamu zveParabolic Conic
Chikamu checonic chikamu chenzvimbo yeconic yakachekwa neplane. Maumbirwo ejometri ezvikamu zveconic anosanganisira madenderedzwa, maellipses, maparabolas, uye mahyperbolas. Muchinyorwa chino, tichatarisa paparabola, imwe yemhando dzakajairika dzezvikamu zveconic dzinowanikwa munzvimbo dzakasiyana dzesainzi, kunyanya masvomhu nefizikisi. Parabola inogona kutsanangurwa seseti yemapoinzi akaenzana kubva panzvimbo yakatarwa (focus) uye mutsetse wakatarwa (directrix).
Tsanangudzo yeParabola
Kuti unzwisise zvakawanda nezve parabola, zvakakosha kunzwisisa zvinhu zvakakosha zve parabola, zvinoti:
1. Vertex (Peak): Nzvimbo inoshanduka parabola apo parabola inochinja curve.
2. Kutarisa: Nzvimbo yakatarwa mudenderedzwa inoshandiswa kutsanangura parabola.
3. Directrix: Mutsetse wakatarwa uri mudenderedzwa unoshandiswa kutsanangura parabola.
4. Axis of Symmetry: Mutsetse unopfuura nepakati pechinhu chakatarisana nemusoro, woparadzanisa parabola kuita zvikamu zviviri zvakafanana.
Equation ye parabola ine vertex iri pakutanga (0,0) inogona kunyorwa nenzira mbiri:
– Horizontal Parabola: \( y^2 = 4ax \)
– Parabola Yakamira: \( x^2 = 4ay \)
apo \(a\) iri chinhambwe kubva pamucheto kusvika pakatarisana.
Mibvunzo yemuenzaniso nehurukuro
Inotevera mienzaniso yakati wandei yemibvunzo uye hurukuro dzayo dzine chekuita nemaparabolas.
Muenzaniso Mubvunzo 1
Mubvunzo:
Sarudza equation ye parabola ine vertex pakutanga (0,0) uye focus panzvimbo (3,0).
Kukurukurirana:
Kubva pamubvunzo, tinoona kuti chinangwa cheparabola chiri pa (3,0). Sezvo chinangwa chiri pa x-axis yakanaka, tinoziva kuti parabola inofanira kunge yakatwasuka.
Pakutsanangura parabola nenzira yakatambanuka, tinoshandisa general equation \( y^2 = 4ax \).
Sezvo chinangwa chiri pa (3,0), saka \(a = 3\).
Saka, equation ye parabola ndeiyi:
\[ y^2 = 4 \cdot 3 \cdot x \]
\[ y^2 = 12x \]
Muenzaniso Mubvunzo 2
Mubvunzo:
Sarudza equation ye parabola ine vertex pakutanga (0,0) uye directrix x = -4.
Kukurukurirana:
Directrix yeparabola ndiyo mutsetse wakagadzika uri kure zvakanyanya kubva pavertex, wakatarisana nefocus. Saka, kana directrix iri x = -4, saka focus iri pa (4,0).
Zvakare, izvi zvinoratidza kuti parabola yakatwasuka.
Kureba kubva pakona kusvika pakatarisana, \(a = 4\).
Equation ye parabola ndeiyi:
\[ y^2 = 4 \cdot 4 \cdot x \]
\[ y^2 = 16x \]
Muenzaniso Mubvunzo 3
Mubvunzo:
Kupiwa parabola ine equation \( x^2 = 8y \). Sarudza macoordinates e vertex, focus, uye directrix equation.
Kukurukurirana:
Kubva muequation \(x^2 = 8y\), zvinoonekwa kuti iyi i parabola yakamira.
Kuti tiwane parabola yechimiro \( x^2 = 4ay \), tinogona kuenzanisa:
\[ 4a = 8 \]
\[ a = 2 \]
Izvi zvinoratidza kuti chinhambwe kubva pakakwirira kusvika pakatarisana i2.
– Makoroniti ePeak: Sezvo pasina kuchinja, peak inoramba iri pakutanga (0, 0).
– Kutarisa: Kutarisa kuri padivi pe y-axis yakanaka iri kure ne a kubva pa vertex, kureva (0, 2).
– Directrix: Directrix ndiyo mutsetse we y = -a, saka directrix ndiyo y = -2.
Muenzaniso Mubvunzo 4
Mubvunzo:
Sarudza equation ye parabola ine focus panzvimbo (0, -2) uye vertex panzvimbo (0, 0).
Kukurukurirana:
Dambudziko iri rinoratidza kuti parabola yakamira uye inoderera (nekuti chinangwa chiri pasi pe vertex).
Pamufananidzo wakamira wakatarisa pasi, chimiro chakajairika ndechekuti \( x^2 = -4ay \).
Kureba kubva pakona kusvika pakatarisana, \( a = 2 \).
Saka, equation ye parabola ndeiyi:
\[ x^2 = -4 \cdot 2 \cdot y \]
\[ x^2 = -8y \]
Muenzaniso Mubvunzo 5
Mubvunzo:
Parabola ine equation \( y^2 + 4y – 4x + 20 = 0 \). Sarudza macoordinates e vertex yayo, focus, uye directrix.
Kukurukurirana:
Danho 1: Chinja chimiro che parabola equation kuita chimiro chakajairwa.
Tanga nekunyora patsva equation:
\[ y^2 + 4y – 4x + 20 = 0 \]
\[ y^2 + 4y = 4x – 20 \]
Danho rechipiri: Pedzisa sikweya yakakwana yechikamu che \(y\):
\[ y^2 + 4y + 4 = 4x – 20 + 4 \]
\[ (y + 2)^2 = 4x – 16 \]
\[ (y + 2)^2 = 4(x – 4) \]
Danho rechitatu: Enzanisa nechimiro chakajairika \( (y – k)^2 = 4a(x – h) \). Muchiitiko ichi, \(a = 1\), \(k = -2\), uye \(h = 4\).
– Nzvimbo dzepamusoro-soro: (4, -2)
– Kutarisa: Sezvo \(a = 1\), daro rayo kubva pakona iri 1 unit. Kutarisa kuri (4+1, -2) = (5, -2).
– Directrix: Mutsetse wakamira unopfuura nepakati pe \( x = h – a = 4 – 1 = 3 \). Saka, directrix ndi \( x = 3 \).
Nekunzwisisa mhando dzakasiyana dzematambudziko uye nzira dzawo dzekugadzirisa, kunzwisisa kwako maparabola kuchavandudzika. Dzidzira matambudziko ane maumbirwo akasiyana-siyana uye magadzirirwo kuti usimbise pfungwa iyi. Maparabola haasi pfungwa yemasvomhu chete asiwo ane mashandisirwo akawanda mufizikisi neinjiniya, kusanganisira nzira dzinoshandiswa nemaprojectile uye parabolic reflectors mumasisitimu ekutaurirana. Kana ukanyanya kudzidzira, ndipo paunonyanya kugona nyaya iyi.
