Mibvunzo yeMienzaniso neKukurukurirana kweZvikamu zveParabolic Conic
Zvikamu zveConic inyaya inokosha mu geometry, inosanganisira maumbirwo akasiyana-siyana akadai semadenderedzwa, ma ellipses, ma hyperbolas, uye ma parabolas. Chimwe chezvimiro zvinonyanya kutaurwa nezvacho uye chinowanzo kurukurwa i parabola. Ma parabolas ane mashandisirwo akawanda mumasvomhu edzidziso uye muhupenyu hwezuva nezuva, senge mukugadzira ndiro dzesatellite uye magirazi emotokari.
Kunzwisisa Parabola
Parabola inogona kutsanangurwa senzvimbo yemapoinzi akaenzana kubva panzvimbo yakatarwa inonzi focus uye mutsetse wakagadzika unonzi directrix. Kana tikafunga nezve parabola muCartesian coordinate system, zvino chinangwa chiri pa x- kana y-axis, zvichienderana nekwakatangira parabola.
Kazhinji, maequations akajairika eparabola ndeaya:
– \( y^2 = 4ax \) ye parabola yakatarisa kurudyi kana kuruboshwe.
– \( x^2 = 4ay \) ye parabola yakatarisa kumusoro kana pasi.
Muenzaniso weMatambudziko eChikamu cheConic cheParabolic
Heino mimwe mienzaniso yemibvunzo ine chekuita nemaparabola nehurukuro dzayo.
Muenzaniso Mubvunzo 1: Kusarudza Chinangwa neDirectrix
Mubvunzo:
Zvichienderana ne equation ye parabola \( y^2 = 8x \). Sarudza ma coordinates e focus uye equation ye directrix.
Kukurukurirana:
Kubva muequation \( y^2 = 8x \), tinogona kunyora kuti parabola iyi ine chimiro \( y^2 = 4ax \) na \( 4a = 8 \) kuitira kuti \( a = 2 \).
– Makonekita ekutarisa: Chinhu chinotariswa neparabola chinonongedzera kurudyi (\( y^2 = 4ax \)) chiri panzvimbo \((a, 0)\) kana \((2, 0)\).
– Muenzaniso weDirectrix: Muenzaniso weDirectrix weparabola iyi mutsetse wakamira une equation \( x = -a \) kana \( x = -2 \).
Saka, ma "focus coordinates" e "parabola" \( y^2 = 8x \) ndi \((2, 0)\) uye "directrix equation" ndi \( x = -2 \).
Muenzaniso Mubvunzo 2: Kuona Equation yeParabola kubva kuFocus neDirectrix
Mubvunzo:
Chinangwa cheparabola ndi \( (3, 0) \) uye directrix ndi \( x = -3 \). Sarudza equation yeparabola.
Kukurukurirana:
Nekuziva focus \( (3, 0) \) uye directrix \( x = -3 \), tinogona kuona kukosha kwe \( a \) kubva pahukama huripo pakati pefocus nedirectrix.
– Chinhambwe kubva pakutarisa kusvika pa y-axis (0) ndi \( 3 \).
– Izvi zvinoreva kuti, daro riri pakati pefocus nedirectrix ndere \( 2a = 3 + 3\), saka \( 2a = 6 \), wozoti \( a = 3 \).
Chimiro chakajairika che parabola yakatarisa kurudyi ndechekuti \( y^2 = 4ax \).
Na \( a = 3 \), tinoitsiva ne parabola equation:
\[ y^2 = 4(3)x \]
\[ y^2 = 12x \]
Saka, equation ye parabola ine chinangwa chiri \( (3, 0) \) uye ine directrix iri \( x = -3 \) ndiyo \( y^2 = 12x \).
Muenzaniso Mubvunzo 3: Kuverenga nzvimbo dzinosangana neCoordinate Axes
Mubvunzo:
Sarudza nzvimbo inopindirana parabola \( y^2 = -16x \) ne coordinate axes.
Kukurukurirana:
Kuti tiwane poindi yekusangana ne x-axis, tinoisa \( y = 0 \) mu equation ye parabola towana kukosha kwe \( x \).
\[ y^2 = -16x \]
Kana \( y = 0 \):
\[ 0 = -16x \]
\[x = 0 \]
Saka, poindi yekusangana ne x-axis ndi \( (0, 0) \).
Kuti tiwane poindi yekusangana ne y-axis, tinoisa \( x = 0 \) tobva tawana kukosha kwe \( y \).
\[ y^2 = -16x \]
Kana \( x = 0 \):
\[ y^2 = -16(0) \]
\[ y^2 = 0 \]
\[ y = 0 \]
Saka, poindi yekusangana ne y-axis zvakare \( (0, 0) \).
Saka, parabola \( y^2 = -16x \) inongosangana chete ne coordinate axes panzvimbo \((0, 0) \).
Muenzaniso Mubvunzo 4: Kudhirowa Parabola
Mubvunzo:
Dhirowa parabola ine equation \( y^2 = -4x \).
Kukurukurirana:
Kuti tigadzire parabola, tinofanira kuziva rumwe ruzivo rwakakosha:
– Parabola iyi yakatarisana nekuruboshwe nekuti x coefficient iri negative.
– Kukosha kwe \( 4a = -4 \) kuitira kuti \( a = -1 \).
Kubva pano, tinogona kunyora:
– Chinangwa che parabola \( (-1, 0) \)
– Directrix \( x = 1 \)
Pakudhirowa, tinogona kutsanangura mamwe mapoinzi ekubatsira:
– Kana \( y = 2 \), \( x = -(\frac{4 \times 2^2}{4}) = -1 \)
– Kana \( y = -2 \), \( x = -(\frac{4 \times (-2)^2}{4}) = -1 \)
Kushandisa mapoinzi akaita se \((0, 0)\), \((-1, 2)\), uye \((-1, -2)\) kuchabatsira pakudhirowa parabola.
Mhedziso
Kunzwisisa zvikamu zve conic, kunyanya parabolas, hakungokoshi chete muzvidzidzo asiwo kune mashandisirwo akawanda anoshanda. Kuburikidza nematambudziko aya emuenzaniso nehurukuro, vaverengi vanotarisirwa kuwana kunzwisisa kwakadzama kwehunhu uye kuongororwa kwema parabolas. Nekuramba vachidzidzira, kunzwisisa uku kuchawedzera uye kuchabatsira kugadzirisa matambudziko ane ma parabolas.
